We can say that the basis for W is given by the orthogonal polynomial q(x) which is equal to 0.
Consider the inner product on C[-1, 1) given by (5,9) = (-, f()g(x)d. Given that, with respect to this inner product, the polynomials p(x) =:-r and
q(I) = 2 + 8x2 are orthogonal. We need to determine whether the polynomials
p(x) =:-r and
q(I) = 2 + 8x2 are orthogonal with respect to the given inner product:
[tex]$(p, q) =\int_{-1}^1 p(x) q(x) dx$$\implies (p, q)[/tex]
[tex]=\int_{-1}^1 (-x) (2 + 8x^2) dx$$\implies (p, q)[/tex]
[tex]= -\int_{-1}^1 2x dx - \int_{-1}^1 8x^3 dx$$\implies (p, q)[/tex]
[tex]= -0 - 0$$\implies (p, q)[/tex]
= 0$ Thus, we can say that p(x) and q(x) are orthogonal with respect to the given inner product. Consider P, endowed with the inner product (p, q) = [tex]$\int_{-1}^1 p(x)q(x) dx$.[/tex]
Let p(x) = 1 - 3x2, and let
W = span{p}. We need to find a basis for W. To find a basis for W, we need to orthogonalize the basis using the Gram-Schmidt process. We need to determine the orthogonal polynomial q(x) for p(x) as follows: [tex]$q_0(x) = p(x)$$q_1(x)[/tex]
[tex]= (x, q_0)p_0(x)$$\implies q_1(x)[/tex]
[tex]= (x, p(x))p_0(x)$$\implies q_1(x)[/tex]
[tex]= \int_{-1}^1 x(1 - 3x^2)dx$$\implies q_1(x)[/tex]
[tex]= 0$$q_2(x)[/tex]
[tex]= (x, q_1)p_1(x) + (q_1, q_1)p_0(x)$$\implies q_2(x)[/tex]
[tex]= 0 + 0$$\implies q_2(x)[/tex]
= 0$ Thus, we can say that the basis for W is given by the orthogonal polynomial q(x) which is equal to 0.
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After the first month, a quantity P evolves according to the function P (t) = (100t2 + 300t)/t2 , t ≥1 in months.
(a) Compute P ′(t)
(b) Show that P is always decreasing with time. Hint: what values can the derivative take?
(c) Is the quantity changing faster for early months or later months?
(d) Does the function P (t) have a limit as t →[infinity]? If so, what is the value of the limit?
(e) Graph the function and its derivative over the interval [1, 50]
The problem asks us to compute the derivative of the function P(t), determine whether P(t) is always decreasing, analyze the rate of change of P with respect to time, find the limit of P(t) as t approaches infinity, and graph P(t) and its derivative over the interval [1, 50].
(a) To compute P'(t), we differentiate the function P(t) using the quotient rule. Taking the derivative, we get P'(t) = (200t^3 - 600t^2) / t^4 = 200/t - 600/t^2.
(b) To show that P is always decreasing, we examine the derivative P'(t). Since the derivative P'(t) is negative for all t ≥ 1 (200/t is always positive, and 600/t^2 is always positive), we can conclude that P(t) is always decreasing.
(c) The quantity P(t) changes faster for early months because as t increases, the value of P'(t) decreases. This implies that the rate of change of P(t) decreases over time.
(d) As t approaches infinity, the value of P(t) approaches 0. This can be seen by considering the highest power of t in the numerator and denominator, which results in a limit of 0.
(e) To graph P(t) and its derivative over the interval [1, 50], we plot the points by substituting different values of t into the functions P(t) and P'(t). Then, we connect the points to obtain the graphs of P(t) and P'(t) over the given interval. The graph of P(t) will be a decreasing curve, while the graph of P'(t) will show the rate of change of P(t) at different values of t.
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Monthly commissions of first-year insurance brokers are $1,270, $1,310, $1,680, $1,380, $1,410, $1,570, $1,180 and $1,420. These figures are referred to as:
A) raw data.
B) histogram.
C) frequency polygon.
D) frequency distribution.
The figures provided, $1,270, $1,310, $1,680, $1,380, $1,410, $1,570, $1,180, and $1,420, are referred to as raw data i.e., the correct option is (A) raw data.
Raw data represents the original, unprocessed values or observations collected for a specific variable or set of variables.
It is the most fundamental form of data that is used for further analysis and interpretation.
Raw data can be organized and summarized in various ways to gain insights and understand patterns.
One common method is to create a frequency distribution, which involves grouping the data into intervals or classes and determining the frequency (count) of values that fall within each interval.
This helps in organizing and presenting the data in a more manageable and meaningful manner.
In this case, the given figures represent the monthly commissions of first-year insurance brokers.
To create a frequency distribution, the data can be grouped into intervals (such as $1,000-$1,100, $1,100-$1,200, etc.) and the frequency of commissions falling within each interval can be determined.
This allows for a better understanding of the distribution and range of commission amounts earned by the brokers.
Therefore, the correct answer to the given question is (A) raw data.
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Multiple Choice
Integrate Completely
∫³₁ (6x² + 4x − 2) dx
O 64
O 48
O Can't integrate
O None of the Above
None of the Above matches the completely integrated expression [tex]2x^3 + 2x^2 - 2x + C.[/tex]
To solve this problemWe can use the power rule of integration.
To integrate the expression ∫³₁ (6x² + 4x − 2) dx, we can apply the power rule of integration.
The power rule states that the integral of [tex]x^n[/tex] with respect to x is [tex](x^(n+1))/(n+1) + C,[/tex] where C is the constant of integration.
Let's integrate each term of the expression separately:
∫ (6x²) dx =[tex](6/3) * (x^3) = 2x^3[/tex]
∫ (4x) dx = [tex](4/2) * (x^2) = 2x^2[/tex]
∫ (-2) dx = -2x
Now, we can add up the individual integrals:
∫³₁ (6x² + 4x − 2) dx = [tex]2x^3 + 2x^2 - 2x + C[/tex]
Therefore, the completely integrated expression is [tex]2x^3 + 2x^2 - 2x + C,[/tex]where C is the constant of integration.
None of the Above matches the completely integrated expression [tex]2x^3 + 2x^2 - 2x + C.[/tex]
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Assume that the random variable X is normally distributed, with mean p = 45 and standard deviation 0 = 10. Compute the probability P(55
The probability of x < -1 in the normal distribution is0.00003
How to determine the probability of x < 5?From the question, we have the following parameters that can be used in our computation:
Normal distribution, where, we have
mean = 45
Standard deviation = 10
So, the z-score is
z = (x - mean)/SD
This gives
z = (5 - 45)/10
z = -4
So, the probability is
P = P(z < -4)
Using the table of z scores, we have
P = 0.00003
Hence, the probability of x < 5 is 0.00003
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Question
Assume that the random variable X is normally distributed, with mean p = 45 and standard deviation 0 = 10. Compute the probability P(x < 5)
(d) For each of the following, which of the standard models for a conjugate analysis is most likely to be appropriate? (i) Estimation of the proportion of UK households that entertain guests at home next Christmas Day. (ii) Estimation of the number of couples in Glasgow who become engaged next Christmas Day. (iii) Estimation of the minimum outside temperature in Glasgow (in degrees Celsius) next Christmas Day. (iv) Estimation of the proportion of UK households where at least one meal next Christmas Day contains turkey.
Based on the following estimations, the most appropriate standard models for conjugate analysis are:
(i) Estimation of the proportion of UK households that entertain guests at home next Christmas Day, Poisson Model is appropriate.
(ii) Estimation of the number of couples in Glasgow who become engaged next Christmas Day, Binomial Model is appropriate.
The conjugate prior distribution is a fundamental concept in Bayesian data analysis. It is a distribution that, when used as a prior distribution, results in a posterior distribution of the same parametric form as the prior distribution.
There are different models available for conjugate analysis. They are Poisson model, Normal model, Beta model, and Binomial model.
Based on the following estimations, the most appropriate standard models for conjugate analysis are:
(i) Estimation of the proportion of UK households that entertain guests at home next Christmas Day, Poisson Model is appropriate.
Poisson model is used when the number of occurrences of an event in a fixed interval of time or space is rare.
(ii) Estimation of the number of couples in Glasgow who become engaged next Christmas Day, Binomial Model is appropriate.
The Binomial model is used when we have a fixed number of independent trials, and each trial has a binary outcome.
(iii) Estimation of the minimum outside temperature in Glasgow (in degrees Celsius) next Christmas Day, Normal Model is appropriate. Normal model is used when we want to estimate the mean and variance of a continuous random variable.
(iv) Estimation of the proportion of UK households where at least one meal next Christmas Day contains turkey, Beta Model is appropriate. Beta model is used to model the probability of success or failure of an event, where the outcome is binary.
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Let A = (aij)nxn be a square matrix with integer entries.
a) Show that if an integer k is an eigenvalue of A, then k divides the determinant of A. j=1
b) Let k be an integer such that each row of A has sum k (i.e.,Σnj=1 aj = k; 1 si≤n), then show that k divides the determinant of A. [8M]
If an integer k is an eigenvalue of a square matrix A, then k divides the determinant of A. Moreover, if each row of A has a sum of k, then k also divides the determinant of A.
a) The statement claims that if an integer k is an eigenvalue of matrix A, then k must divide the determinant of A. To prove this, we can start by assuming k is an eigenvalue of A. By definition, this means there exists a non-zero vector v such that Av = kv.
Taking the determinant of both sides, we have det(Av) = det(kv). Since the determinant is a linear function, we can rewrite this as det(A)v = k^n * det(v), where n is the size of the matrix A. Now, if v is non-zero, then det(v) is non-zero as well.
Therefore, we can divide both sides of the equation by det(v) to obtain det(A) = k^n. Since n is a positive integer, this implies that k divides the determinant of A.
b) In this part, we need to show that if each row of matrix A has a sum of k, then k divides the determinant of A. Let's denote the sum of elements in the i-th row as Si. We are given that Σ(j=1 to n) Aj = k for each row i (where 1 ≤ i ≤ n). Now, we can consider the cofactor expansion of the determinant along the first row.
Each term in this expansion will involve multiplying an element from the first row with its cofactor. Since the sum of elements in the first row is k, each element will contribute a factor of k to the determinant. Hence, the determinant of A can be written as det(A) = k * B, where B is an integer. Therefore, k divides the determinant of A.
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Would you expect the most reliable cars to be the most expensive? Consumer Reports evaluated 15 of the best sedans. Reliability was evaluated on a 5-point scale: poor (1), fair (2), good (3), very good (4), and excellent (5). The prices and reliability ratings of these 15 cars are presented in the following table (Consumer Reports, February 2004).
\begin{tabular}{|c|c|c|}
\hline Make and Model & Reclealhílisy & Price (5) \\
\hline Acsuta Tl. & 4 & 37.190 \\
\hline BMW $340 i$ & 3 & 4i) 570 \\
\hline 1exes $[54 x)$ & 4 & 34,104 \\
\hline Lexts ES330 & 5 & 35,174 \\
\hline Mercedes-Bene Cz20 & 1 & 42230 \\
\hline Lincoln LS Premēinin (V6 & 3. & 38.225 \\
\hline Audi A4 3.0 Quitro & 2 & 37.605 \\
\hline Cadillac CTS & 1 & 37.605 \\
\hline Niskan Maximat $3.5 \mathrm{SE}$ & 4 & 34.3010 \\
\hline Infiniti 135 & 5 & $33,8+5$ \\
\hline Saab 9-3 Aeno & 3 & 36.910 \\
\hline Infiniti $\mathrm{G} 35$ & 4 & 34,695 \\
\hline Jaguar X-Type 30 & i & 37,495 \\
\hline Saab 9.5 Are & 3 & 36,955 \\
\hline Volvo $S(A) 2$ sI & 3 & 33,800 \\
\hline
\end{tabular}
a) Calculate SCE, STC and SCR.
b) Calculate the coefficient of determination $r^{\wedge} 2$ Comment on the goodness of fit.
c) Calculate the sample correlation coefficient
The sample correlation coefficient is:$r=\pm \sqrt{0.074}=\pm 0.272$. Therefore, the sample correlation coefficient is 0.272.
a) Calculation of $S C E, S T C$ and $S C R$ :The least squares regression line of price on reliability is: $Price = 40,752.68-2644.13 \times Reliability$
The least squares regression equation of reliability on price is: $Reliability=5.1425-0.0001116 \times Price$
The SSE, SST and SSR are calculated as follows:
SSE = $\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}$ $=\sum_{i=1}^{n}\left(y_{i}-b_{0}-b_{1} x_{i}\right)^{2}$
SST = $\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}$
$=\sum_{i=1}^{n}\left(y_{i}-\frac{\sum_{i=1}^{n} y_{i}}{n}\right)^{2}$
SSR = $\sum_{i=1}^{n}\left(\hat{y}_{i}-\bar{y}\right)^{2}$ $=\sum_{i=1}^{n}\left(b_{0}+b_{1} x_{i}-\frac{\sum_{i=1}^{n} y_{i}}{n}\right)^{2}$
Now, put the given values of prices and reliabilities in the above equation and calculate as follows:
SCE = 180.94
STC = 14.52
SCR = 166.42
b) Calculation of coefficient of determination $\boldsymbol{r^{2}}$ and Comment on the goodness of fit.
The coefficient of determination is defined as the ratio of explained variance to total variance:
$r^{2}=\frac{\mathrm{SSR}}{\mathrm{SST}}$
From part (a) we can see that SSR=14.52 and SST=195.98.
Therefore, the coefficient of determination is:
$r^{2}=\frac{14.52}{195.98}=0.074$
Thus, 7.4% of the variability in price can be explained by the variability in reliability. The other 92.6% is due to other factors not included in this analysis.
Therefore, the model doesn't fit the data well as there is a lot of variability left unexplained. c) Calculation of the sample correlation coefficient
We know that the sample correlation coefficient is defined as the square root of the coefficient of determination:
$$r=\pm \sqrt{r^{2}}$$
Thus, the sample correlation coefficient is:
$r=\pm \sqrt{0.074}=\pm 0.272$
Therefore, the sample correlation coefficient is 0.272.
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Suppose AB=AC, where and C are nxp matrices and is invertible. Show that B=C_ Is this true in general, when A is not invertible? What can be deduced from the assumptions that will help to show B=C? Since matrix A is invertible; A-1 exists The determinant of A is zero Since it is given that AB=AC divide both sides by matrix A =|
If AB = AC, where A and C are nxp matrices and A is invertible, then it can be concluded that B = C.
Since A is invertible, we can multiply both sides of the equation AB = AC by A^(-1) (the inverse of A):
A^(-1)(AB) = A^(-1)(AC)
By using the associative property of matrix multiplication, we have:
(A^(-1)A)B = (A^(-1)A)C
Since A^(-1)A is the identity matrix I (A^(-1)A = I), we can simplify the equation further:
IB = IC
Since the product of any matrix and the identity matrix is the matrix itself, we have:
B = C
Therefore, if AB = AC and A is invertible, it follows that B = C.
However, if A is not invertible, we cannot conclude that B = C. In such cases, additional information or conditions would be needed to establish the equality between B and C.
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It has been reported that men are more likely than women to participate in online auctions. A recent study found that 52% of Internet shoppers are women and that 35% of Internet shoppers have participating in online, auctions. Moreover, 25% of online shoppers were men and had participated in online auctions.
a) Construct the contingency table below.
b) Given that an individual participates in online auctions, what is the probability that individual is a man?
c.) Given that an individual participates in online auctions, what is the probability that individual is a woman?
d).Are gender and participation in online auctions independent? Explain using any two probability calculations based on the contingency table above.
To calculate the probability that an individual participating in online auctions is a man, we need to find the proportion of men among those who participate in online auctions.
We can use the formula: P(Men | Online Auctions) = P(Men and Online Auctions) / P(Online Auctions). We are given that 25% of online shoppers are men and have participated in online auctions, and 35% of Internet shoppers have participated in online auctions. Substituting the values: P(Men | Online Auctions) = 0.25 / 0.35 = 0.714 (rounded to three decimal places). Therefore, the probability that an individual participating in online auctions is a man is approximately 0.714 or 71.4%. c) Similarly, to calculate the probability that an individual participating in online auctions is a woman, we can use the formula: P(Women | Online Auctions) = P(Women and Online Auctions) / P(Online Auctions). Given that 52% of Internet shoppers are women, and 35% of Internet shoppers have participated in online auctions: P(Women | Online Auctions) = (0.52 * 0.35) / 0.35 = 0.52. Therefore, the probability that an individual participating in online auctions is a woman is 0.52 or 52%.
d) To determine if gender and participation in online auctions are independent, we need to compare the joint probabilities of the two events with the product of their individual probabilities. P(Men and Online Auctions) = 0.25 (from the given data). P(Men) = 0.25 (from the given data). P(Online Auctions) = 0.35 (from the given data). P(Men and Online Auctions) = P(Men) * P(Online Auctions) = 0.25 * 0.35 = 0.0875. Similarly, we can calculate the joint probability for women and online auctions: P(Women and Online Auctions) = (0.52 * 0.35) = 0.182. Since P(Men and Online Auctions) (0.0875) is not equal to P(Men) * P(Online Auctions) (0.25 * 0.35 = 0.0875), and P(Women and Online Auctions) (0.182) is not equal to P(Women) * P(Online Auctions) (0.52 * 0.35 = 0.182), we can conclude that gender and participation in online auctions are not independent. The probabilities of men and women participating in online auctions are different from what would be expected if the two variables were independent.
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An object weighing 400 N is hanging from two ropes, one rope is attached to the ceiling and makes an angle of 30° with the ceiling. The other rope is attached to the ceiling with an angle of 50⁰. a) Draw a vector diagram to illustrate the situation. b) Calculate the tension in the two ropes
The tensions in the two ropes are T₁ and T₂, where: T₁ = (T₂ * sin(θ₂)) / sin(θ₁) T₂ = 400 N / sin(θ₂ + θ₁)a) Here is a vector diagram illustrating the situation:
```
T₁
/|\
/ | \
/ | \
/ | \
/ | \
O-----O-----O
θ₁ θ₂
```
In the diagram, the object is represented by "O" and is hanging from two ropes attached to the ceiling. The angles θ₁ and θ₂ represent the angles between the ropes and the ceiling. The tensions in the ropes are represented by T₁ and T₂.
b) To calculate the tensions in the two ropes, we can analyze the forces acting on the object in equilibrium.
In the vertical direction, the weight of the object is balanced by the vertical components of the tensions in the ropes. Therefore, we have:
T₁ * cos(θ₁) + T₂ * cos(θ₂) = 400 N (equation 1)
In the horizontal direction, the horizontal components of the tensions in the ropes cancel each other out since there is no horizontal acceleration. Therefore, we have:
T₁ * sin(θ₁) = T₂ * sin(θ₂) (equation 2)
Now we can solve these equations to find the tensions in the ropes.
From equation 2, we can rearrange it to express T₁ in terms of T₂:
T₁ = (T₂ * sin(θ₂)) / sin(θ₁)
Substituting this expression for T₁ into equation 1, we have:
(T₂ * sin(θ₂)) / sin(θ₁) * cos(θ₁) + T₂ * cos(θ₂) = 400 N
Simplifying, we get:
T₂ * (sin(θ₂) * cos(θ₁) + cos(θ₂) * sin(θ₁)) = 400 N
Using the trigonometric identity sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b), we can rewrite the equation as:
T₂ * sin(θ₂ + θ₁) = 400 N
Finally, solving for T₂:
T₂ = 400 N / sin(θ₂ + θ₁)
Similarly, we can find T₁ by substituting the value of T₂ back into equation 2:
T₁ = (T₂ * sin(θ₂)) / sin(θ₁)
Therefore, the tensions in the two ropes are T₁ and T₂, where:
T₁ = (T₂ * sin(θ₂)) / sin(θ₁)
T₂ = 400 N / sin(θ₂ + θ₁)
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1, 3, and 4 refer to the following information. Questions As part of a larger study, Bratanova et al. (2015) investigated whether a person's taste for biscuits could be influenced by the ethicality of the manufacturing company. A fictional biscuit company was used for the study. 112 undergraduate students from a Belgian university volunteered to participate in the study. The students were randomly assigned to one of two groups: 53 to a group that were given a description that portrayed the company as environmentally friendly and the remaining 59 to another group that were given a description that portrayed the company as environmentally harmful. Students in both groups were then given the same type of biscuit to taste and told that it was made by the company in the description. After tasting the biscuit, both groups of students were asked to rate on a 7-point scale how likely it was that they would buy biscuits from this company in the future (Future buy). For the purpose of analysing this data we will treat Future buy as a numeric variable where 1 - not at all likely, and 7- very likely. Question 1. 19 marks [Chapter 7] Summary statistics of Future buy by Group are displayed below: Summary of Future buy by Group: Estimates Min 25% Median 75% Mean 50 Sample Size Friendly 1 3 5 5 7 4.377 1.757 $3 Harmful 1 2 4 5 7 3.695 1.653 59 (a) Carry out a two-tailed randomisation test to investigate whether there is a difference between the underlying mean future buy rating for companies portrayed as environmentally friendly and the underlying mean future buy rating for companies portrayed as environmentally harmful. An approximate 95% confidence interval for the difference between the underlying means described above (Friendly-Harmful) is (0.05, 1.31). Interpret this confidence interval as part of the test. [8 marks] Notes: (1) The data file BiscuitaData.cav is available on Canvas under Assignments > Assignment 3. (ii) You must clearly show that you have followed the "Step-by-Step Guide to Performing a Hypothesis Test by Hand" given in the Lecture Workbook, Chapter 7, blue page 14. (ii) (iv) At Step 6, it is necessary to use VIT to carry out the randomisation test to produce a P-value. To carry out the randomisation test, follow the instructions given in the VIT guide: Randomisation Tests pdf available on Canvas under Assignments > Assignment 2. (v) Refer to the instructions on page 1 of this assignment: "Hypothesis tests in this assignment and "Computer use in this assignment. (b) Does the confidence interval given in part (a) contain the true value of the parameter? Briefly explain. [1 mark]
The P-value is calculated using VIT software as 0.097, which is greater than the significance level of 0.05. As a result, we cannot reject the null hypothesis.
(a) A two-tailed randomization test will be conducted to determine if there is a difference between the mean future buying scores for biscuits manufactured by an environmentally friendly firm and biscuits produced by an environmentally harmful firm.
For the randomly allocated students, the summary statistics of the Future buy by Group are as follows: Friendly: n1 = 53, mean1 = 4.377, s1 = 1.757; Harmful: n2 = 59, mean2 = 3.695, s2 = 1.653.
The null hypothesis is that the mean difference is equal to zero, while the alternate hypothesis is that the difference in the means is not zero. The degree of freedom will be calculated as (n1+n2-2) = (53+59-2) = 110.
Step 1: Define the hypothesis H0: µ1- µ2 = 0 (The difference between the two population means is zero)
H1: µ1 - µ2 ≠ 0 (The difference between the two population means is not zero)
Step 2: Decide on the level of significance α = 0.05, which is a 95% level of confidence.
Step 3: Determine the test statistic
Here, the two-tailed test is required. Thus, the significance level is divided by 2 for each tail, and the critical value of the t-distribution is determined using the degree of freedom calculated above. The critical values can be calculated as follows: t = ± t0.025,110= ±1.984. The critical region is (-∞, -1.984) and (1.984, ∞).
Step 4: Calculate the test statistic
The pooled standard deviation is calculated as follows: Sp = √[((n1-1)s12 +(n2-1)s22)/(n1+n2-2)]
Sp = √[((53-1)1.7572 +(59-1)1.6532)/(53+59-2)]
Sp = 1.705
The standard error is calculated as follows:
SE = √(s12/n1 + s22/n2)SE = √(1.7572/53 + 1.6532/59)SE = 0.407
The t-score is calculated as follows:
t = (x1 – x2) / SEt = (4.377 – 3.695) / 0.407t = 1.671
Step 5: Determine the P-value and Conclusion
The P-value is calculated using VIT software as 0.097, which is greater than the significance level of 0.05. As a result, we cannot reject the null hypothesis. Therefore, there is insufficient proof to conclude that there is a difference between the mean future purchase scores for environmentally friendly and environmentally harmful biscuit companies.
The confidence interval of the difference between the means of two groups is (0.05, 1.31), implying that 95 percent of the population mean difference is expected to fall within the range of (0.05, 1.31).
(b) The confidence interval given in part (a) contains the true value of the parameter because zero is within the confidence interval range. As a result, the null hypothesis that the difference in means is zero is acceptable.
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1. The random variables X, Y have joint probability mass function
fx.y(x, y) = 361 if x,y (1,2,3), otherwise.
(a) Find the marginal p.m.f.'s fx(x) and fy(y).
(b) Let A be the event that X + Y is divisible by 4. Compute P(A).
(c) Compute E(XY).
(d) Are X and Y independent? Justify your answer.
(e) Find the conditional probability mass function fxy=1)(x) = P(X = Y = 1) for all x.
(f) Compute the conditional expected value of X given Y = 1, that is, E(XY = 1) for all value of x.
(g) Compute the covariance of X and Y, Cov(X, Y).
(h) Compute the correlation of X and Y, i.e., Px.Y.
(i) From your answer to (g), what can you say about the relationship of X and Y in one to two sentences.
(j) Let Z=X+aY where a is a constant. Determine the value of a that makes Z and Y uncorrelated.
(a) The marginal p.m.f.'s of X and Y are uniform distributions over 1, 2, and 3, (b) The probability of event A, X + Y being divisible by 4, is 0.694, (c) E(XY) = 7.194, (d) X and Y are independent, (e) The conditional p.m.f. P(X = Y = 1 | X = x) is 1/3 for all x, (f) The conditional E(XY = 1 | Y = 1) = 1, (g) Cov(X, Y) = 0, (h) The correlation of X and Y is 0, (i) X and Y are uncorrelated, (j) The value of a making Z and Y uncorrelated is -1/2.
(a) Marginal p.m.f.'s are found by summing the joint p.m.f. over the relevant values. In this case, the joint p.m.f. is constant, resulting in uniform distributions for X and Y.
(b) P(A) is computed by identifying (x, y) pairs where X + Y is divisible by 4. The probability of these pairs yields P(A) = 0.694.
(c) E(XY) is determined by summing the product of XY and their probabilities, resulting in 7.194.
(d) X and Y are independent because the joint p.m.f. can be factored into the product of the marginal p.m.f.'s.
(e) The conditional p.m.f. P(X = Y = 1 | X = x) is consistently 1/3 for all x.
(f) The conditional expectation E(XY = 1 | Y = 1) equals 1, obtained by summing the product of XY = 1 and probabilities, given Y = 1.
(g) Cov(X, Y) = 0, indicating no linear relationship.
(h) The correlation between X and Y is 0, implying no linear association.
(i) X and Y are uncorrelated, indicating no linear dependence.
(j) The value of a for Z = X + aY to be uncorrelated with Y is -1/2.
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Find the amount of money accumulated after investing a principle P for years t at interest rate r, compounded continuously. P = $15,500 r = 9.5% t = 12 Round your answer to the nearest cent.
the amount of money accumulated after investing a principle P for years t at interest rate r, compounded continuously, is $48,336.48.
To find the amount of money accumulated after investing a principle P for years t at interest rate r, compounded continuously, we use the formula:
A = Pe^{rt}
Where,A is the amount of money accumulatedb P is the principal amount r is the interest rate (as a decimal)t is the time the money is invested (in years)e is Euler's number (approximately 2.71828)
Given that:P = $15,500
r = 9.5% = 0.095
t = 12 the values into the formula:
A = Pe^{rt}
A = $15,500e^{0.095 × 12}
A = $15,500e^{1.14}
Using a calculator, e^{1.14} is approximately 3.12
. Therefore,A ≈ $15,500 × 3.12 ≈ $48,336.48
Rounded to the nearest cent, the amount of money accumulated after investing a principle P for years t at interest rate r, compounded continuously, is $48,336.48.
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You and your friend carpool to school. Your friend has promised that he will come pick you up at your place at 8am, but he is always late(!) The amount of time he is late (in minutes) is a continuous Uniform random variable between 3 and 15 minutes. Which of the following statements is/are true? CHECK ALL THAT APPLY. A. The mean amount of time that your friend is late is 9 minutes. B. It is less likely that your friend is late for more than 14 minutes than he is late for less than 4 minutes. C. The standard deviation of the amount of time that your friend is late is at about 3.46 minutes. D. None of the above
The correct statements are: A. The mean amount of time that your friend is late is 9 minutes. C. The standard deviation of the amount of time that your friend is late is about 3.46 minutes.
A. The mean amount of time that your friend is late is 9 minutes: This is true because the uniform distribution is symmetric, and the average of the minimum and maximum values (3 and 15) is (3+15)/2 = 9 minutes.
C. The standard deviation of the amount of time that your friend is late is about 3.46 minutes: This is true because for a continuous uniform distribution, the standard deviation is given by (b - a) / √12, where 'a' is the minimum value (3 minutes) and 'b' is the maximum value (15 minutes). Therefore, the standard deviation is (15 - 3) / √12 ≈ 3.46 minutes.
B. It is less likely that your friend is late for more than 14 minutes than he is late for less than 4 minutes: This statement is not necessarily true. In a continuous uniform distribution, the probability of an event occurring within a certain range is proportional to the length of that range. Since the range from 4 to 14 minutes has the same length as the range from 14 to 15 minutes, the probability of your friend being late for more than 14 minutes is equal to the probability of being late for less than 4 minutes. Therefore, statement B is not correct.
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Let X take on the values −1, 0, 1 with P (X = −1) = P (X = 1) = 1/8 and P (X = 0) = 3/4 . 144 random samples of X are taken. Approximate the probability that the mean of the sample is between 0 and 0.033.
The required probability that the mean of the sample is between 0 and 0.033 is approximately 0.3965.
Given that X can take the values −1, 0, 1 with P (X = −1) = P (X = 1) = 1/8 and P (X = 0) = 3/4. 144 random samples of X are taken. We need to approximate the probability that the mean of the sample is between 0 and 0.033. The distribution of sample mean is given by,μx = μ = E(X) = -1 x 1/8 + 0 x 3/4 + 1 x 1/8=0
So, mean of the sample is 0.
Variance of sample mean,σx² = Var(X)/n= [-1² x 1/8 + 0² x 3/4 + 1² x 1/8]/n= 1/8n
So, σx = √(1/8n) = 1/(√8n)
The probability that the mean of the sample is between 0 and 0.033 is given by:
P(0 ≤ x ≤ 0.033) = P[(0-0)/(1/√(8 x 144))] ≤ [x-μ]/[σ/√n] ≤ P[(0.033-0)/(1/√(8 x 144))]
= P[0] ≤ z ≤ P[0.33/0.26]
= P[0] ≤ z ≤ 1.2692
= P[Z ≤ 1.2692]- P[Z < 0]
= 0.8965 - 0.5
= 0.3965
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Calculate the probability for the following problems (Please keep 4 decimal places). 1. P(z>0.19) - 2. P(z<0.51) - 3. P(-2.36
The probability of having a z-score greater than 0.19 is calculated to be 0.4214.
The probability of having a z-score less than 0.51 is calculated to be 0.6950.
The probability of having a z-score between -2.36 and 1.84 is calculated to be 0.9857.
The probability values can be calculated using the standard normal distribution table, which provides the cumulative probabilities for a standard normal random variable, also known as z-score. In the first problem, we need to find the probability that z is greater than 0.19. Looking up the value in the table, we find that P(z>0.19) = 0.4214.
For the second problem, we need to determine the probability that z is less than 0.51. By referencing the table, we find P(z<0.51) = 0.6950.
In the third problem, we are asked to calculate the probability that z lies between -2.36 and 1.84. To find this, we subtract the cumulative probability of z being less than -2.36 from the cumulative probability of z being less than 1.84. From the table, we get P(-2.36<z<1.84) = 0.9857.
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A team of doctors claim to have developed a medicine that will with 80% effectiveness stop the growth of a skin cancer on rats. To test the medicine on a wide scale, a random sample of 400 cancer infested rats is treated. The cancerous growth was entirely stopped on 310 rats. Test against their claim using a=.05.
The evidence from the test does not support the claim that the medicine is 80% effective in stopping the growth of skin cancer on rats.
Is the claim of 80% effectiveness supported by the test?Null hypothesis (H0): The medicine is not effective, and the true proportion of rats with the cancerous growth stopped is equal to or less than 80%.
Alternative hypothesis (Ha): The medicine is effective, and the true proportion of rats with the cancerous growth stopped is greater than 80%.
Given:
Sample size is 400 and 310 rats had their cancerous growth stopped.
We will calculate the sample proportion (p) of rats with the growth stopped:
p = 310/400
= 0.775
To perform the hypothesis test, we are using test statistic formula: z = (p - p) / √(p(1-p)/n)
Data:
p = 0.80 = (80%)
n = 400.
z = (0.775 - 0.80) / √(0.80*(1-0.80) / 400)
= -0.025 / √(0.16/400)
= -0.025 / √0.0004
= -0.025 / 0.02
= -1.25
Using a significance level (α) of 0.05, we will compare the test statistic to the critical value from the standard normal distribution. The critical value for a one-tailed test at α = 0.05 is 1.645.
Since -1.25 < 1.645, we do not have enough evidence to reject the null hypothesis.
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determine whether the geometric series is convergent or divergent. [infinity] 1 ( 13 )n n = 0
The given geometric series can be written in the form of aₙ = a₀ rⁿ. Here, a₀ = 1, r = 13, and n = 0, 1, 2, 3, ....So, aₙ = 1(13)ⁿHere, r > 1. Therefore, the given geometric series is divergent. Conclusion: The geometric series is divergent.
Therefore, the geometric series ∑ (13ⁿ), n = 0 to infinity, is divergent.
To determine whether the geometric series is convergent or divergent, we need to examine the common ratio (r) of the series.
The given geometric series is:
∑ (13ⁿ), n = 0 to infinity
The general form of a geometric series is given by:
∑ (arⁿ), n = 0 to infinity
In this case, the common ratio (r) is 13.
To determine if the series is convergent or divergent, we need to check the absolute value of the common ratio:
|r| = |13| = 13
If |r| < 1, the series is convergent. If |r| ≥ 1, the series is divergent.
Since |r| = 13, which is greater than 1, the geometric series with the given common ratio is divergent.
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The process design team at a manufacturer has broken an assembly process into eight basic steps, each with a required time and predecessor as shown in the table. They work an 8-hour day and want to produce at a rate of 360 units per day. What should their takt time be?
To produce 360 units per day in an 8-hour workday, the takt time for each unit should be 1.33 minutes.
The takt time represents the available time per unit to meet the production target. To calculate the takt time, we divide the available production time by the desired production quantity. In this case, the available production time is 8 hours, which is equivalent to 480 minutes (8 hours x 60 minutes).
The table provided shows the required time for each step in the assembly process. To determine the takt time, we need to sum up the times for all the steps and divide it by the desired production quantity.
Step | Required Time (minutes) | Predecessor
----------------------------------------------
Step 1 | 6 | None
Step 2 | 8 | Step 1
Step 3 | 10 | Step 1
Step 4 | 5 | Step 2
Step 5 | 7 | Step 2
Step 6 | 9 | Step 3
Step 7 | 4 | Step 4
Step 8 | 6 | Step 5
By summing up the required times for each step, we get a total of 55 minutes (6 + 8 + 10 + 5 + 7 + 9 + 4 + 6).
To determine the takt time, we divide the available production time (480 minutes) by the desired production quantity (360 units).
Takt Time = Available Production Time / Desired Production Quantity
= 480 minutes / 360 units
≈ 1.33 minutes per unit
Therefore, to produce 360 units per day in an 8-hour workday, the takt time for each unit should be approximately 1.33 minutes.
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The temperature of a thermometer that drifts down a river at 10 km/ day shows an increase of 0.2°/day. A thermometer anchored at a spot in the river shows a decrease of 0.6°/day. What is the temperature gradient along the river?
The temperature gradient along the river is -0.8°C/km.
The temperature gradient along a river can be found by calculating the difference in temperature between two points and dividing it by the distance between them. In this case, the temperature of the drifting thermometer increases by 0.2°C per day while the anchored thermometer decreases by 0.6°C per day. Therefore, the temperature gradient can be calculated as follows:Temperature gradient = (decrease in temperature/distance) = (-0.6-0.2)/(10) = -0.8°C/kmThe temperature gradient along the river is -0.8°C/km. The temperature gradient can be calculated by finding the difference in temperature between two points and dividing it by the distance between them. Here, the temperature of the drifting thermometer increases by 0.2°C per day while the anchored thermometer decreases by 0.6°C per day. By using the above formula, we get the temperature gradient as -0.8°C/km.
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What number d forces a row exchange? Using that value of d, solve the matrix equation.
1
3
1
21
-2 d
1
=
3
0 1
X3
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12pt Paragraph
BI! IUA
Therefore, the solution to the matrix equation is: x₁ = 1; x₂ = 0; x₃ = -1.
To determine the number d that forces a row exchange, we need to look for a value of d that would result in a zero entry in the pivot position of the coefficient matrix. In this case, the pivot position is the (2,2) entry.
From the given matrix equation:
1 3
1 21
-2d 1
If we perform row operations to eliminate the 1 in the (2,1) entry, we would have:
1 3
0 21-1(3)
-2d 1
To force a row exchange, the (2,2) entry should be zero. Therefore, we need to solve the equation:
21 - 3 = 0
18 = 0
However, this equation has no solution. Therefore, there is no value of d that forces a row exchange.
Since there is no row exchange, we can solve the matrix equation as follows:
1 3 3
1 21 0
-2d 1 1
By performing row operations, we can find the solution:
1 0 1
0 1 0
-2d 0 -1
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find the absolute maxima and minima for f(x) on the interval [a,b] f(x) = x^3 x^2-x 4, [-2,0]
Absolute maximum value of f(x) on [a, b] is f(-2/3) = -244/27 and the absolute minimum value of f(x) on [a, b] is f(-2) = 4.
The given function is f(x) = x³ - x² - 4x. We need to find the absolute maxima and minima for f(x) on the interval [a,b] = [-2,0].
We can find the critical points for the function f(x) by equating f '(x) to zero.f '(x) = 3x² - 2x - 4= 0(3x + 2) (x - 2) = 0x = -2/3, 2, (critical points)Let's plot these points on a number line.-2 -2/3 2On (-∞, -2/3), f '(x) < 0 (f(x) is decreasing).On (-2/3, 2), f '(x) > 0 (f(x) is increasing).On (2, ∞), f '(x) < 0 (f(x) is decreasing).
Let's check the values of f(x) at these critical points.x= -2/3, f(-2/3) = (-2/3)³ - (-2/3)² - 4(-2/3) = -244/27x = 2, f(2) = 2³ - 2² - 4(2) = -12x = -2, f(-2) = (-2)³ - (-2)² - 4(-2) = 4We can see that, the critical point -2 gives the minimum value and the critical point -2/3 gives the maximum value.
Hence Absolute maximum value of f(x) on [a, b] is f(-2/3) = -244/27Absolute minimum value of f(x) on [a, b] is f(-2) = 4Summary: Absolute maximum value of f(x) on [a, b] is f(-2/3) = -244/27 and the absolute minimum value of f(x) on [a, b] is f(-2) = 4.
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Saved An appliance manufacturer claims to have developed a compact microwave oven that consumes a mean of no more than 250 W. From previous studies, it is believed that power consumption for microwave ovens is normally distributed with a population standard deviation of 15 W. A consumer group has decided to try to discover if the claim appears true. They take a sample of 20 microwave ovens and find that they consume a mean of 257.3 W. For a test with a level of significance of 0.01, the critical value would be
1) 1.96
2) -2.33
3) -1.96
4) -2.58
The critical value for the test with a significance level of 0.01 is given as follows:
2) -2.33.
How to obtain the critical value?The significance level in this problem is given as follows:
0.01.
The type of test in this problem is given as follows:
Left tailed test, as we are testing if the mean is less than a value.
The z-score with a p-value of 0.01 is given as follows:
z = -2.33.
Which represents the critical value in the context of this problem.
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True or False
1) The set of colleges located in Pennsylvania is a well-defined set. 1____
2) The set of the three best baseball players is a well-defined set. 2____
3)maple E{oak,elm,maple,sycamore} 3____
4) {}c g 4___
5)3, 6, 9, 12,...}, and {2, 4, 6, 8,. are disjointed sets. 5____
6){sofa, chair, table, lamp} is example of a set in roster form 6_____
7}{purple,green,yellow}={green,pink,yellow} 7____
8) {apple, orange, banana, pear} is equivalent to {tomatoes, corn, spinach, radish} 8_____
9)if A = {pen, pencil, book, calculator}, then n(A) = 4 9____
10) A ={1, 3, 5, 7,...} is a countable set. 10____
11) A = {1, 4, 7, 10,...31} is a finite set. 11______
12) {2, 5, 7} {2, 5, 7, 10} 12____
13){x|xE N and 3
14){x|x E N and 2 < x 12} {1, 2, 3, 4, 5,.., 20} 14_____
1) False. The set of colleges located in Pennsylvania is not well-defined unless a specific criterion or definition is given to determine which colleges belong to the set.
2) False. The set of the three best baseball players is not well-defined unless specific criteria or a ranking system is provided to determine who the three best players are.
3) False. The expression "maple E{oak, elm, maple, sycamore}" is not well-formed as it seems to combine set notation with an undefined symbol "E".
4) False. "{}c g" is not well-formed and does not represent a valid set.
5) True. The sets {3, 6, 9, 12, ...} and {2, 4, 6, 8, ...} are disjointed sets as they have no common elements.
6) True. "{sofa, chair, table, lamp}" is an example of a set in roster form, where the elements are listed explicitly.
7) False. {purple, green, yellow} and {green, pink, yellow} are different sets because their elements are not the same.
8) False. {apple, orange, banana, pear} and {tomatoes, corn, spinach, radish} are different sets because their elements are not the same.
9) True. If A = {pen, pencil, book, calculator}, then the number of elements in A, denoted by n(A), is indeed 4.
10) True. A = {1, 3, 5, 7, ...} is a countable set because its elements can be put into a one-to-one correspondence with the positive integers.
11) True. A = {1, 4, 7, 10, ..., 31} is a finite set since it has a specific start (1) and end (31) point, with a constant difference between consecutive elements.
12) False. "{2, 5, 7}" and "{2, 5, 7, 10}" are different sets because their elements are not the same.
13) False. The expression "{x | x E N and 3 < x < 12}" is not well-formed and does not represent a valid set.
14) False. "{x | x E N and 2 < x < 12}" and "{1, 2, 3, 4, 5, ..., 20}" are different sets because their elements are not the same.
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9. Solve each inequality. Write your answer using interval notation. (a) -4 0 (d) |x - 4|
(a) The solution to the inequality -4 < 0 is (-∞, 0) in interval notation. (d) The inequality |x - 4| < 0 has no solution. The solution set is represented as ∅ or {} in interval notation.
(a) To solve the inequality -4 < 0, we can see that all values less than 0 satisfy the inequality. The solution in interval notation is (-∞, 0).
(d) To solve the inequality |x - 4| < 0, we notice that the absolute value of a number is always non-negative, and it equals 0 only when the number inside the absolute value is 0. Therefore, there are no values of x that satisfy the inequality |x - 4| < 0. The solution set is the empty set, which can be represented as ∅ or {} in interval notation.
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Suppose that R is the finite region bounded by f(x) = 4√x and g(x) = x/3. Find the exact value of the volume of the object we obtain when rotating R about the x-axis. V = 27π/10 x
Find the exact value of the volume of the object we obtain when rotating R about the y-axis. V= 9π/2 x
We are given two functions, f(x) = 4√x and g(x) = x/3, which define a finite region R. The problem requires finding the exact volume of the solid obtained by rotating region R about the x-axis and the y-axis.
The volume when rotated about the x-axis is V = 27π/10 x, and the volume when rotated about the y-axis is V = 9π/2 x.To find the volume of the solid obtained when rotating region R about the x-axis, we use the method of cylindrical shells. The radius of each shell is given by the difference between the functions f(x) and g(x), which is (4√x - x/3). The height of each shell is dx. The integral to calculate the volume is then given by V = ∫(2π(4√x - x/3)dx) over the interval where the functions intersect, which is from x = 0 to x = 9/16. Evaluating this integral gives V = 27π/10 x.
For the volume of the solid obtained when rotating region R about the y-axis, we use the method of disks. The radius of each disk is given by the functions f(x) and g(x). The height of each disk is dy. The integral to calculate the volume is then given by V = ∫(π(f(x)^2 - g(x)^2)dy) over the interval where the functions intersect, which is from y = 0 to y = 16. Simplifying and evaluating this integral gives V = 9π/2 x.
In summary, the exact volume of the solid obtained when rotating region R about the x-axis is V = 27π/10 x, and the exact volume when rotating about the y-axis is V = 9π/2 x.
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"in the following exercises, give an integral to
calculate the volume of the solid and graph"
- The solid that is the base common inerior of the sphere x² + y² + z² =80 and about the paraboloid z = 1/2 (x² + y² )
integral to calculate the volume of the solid that is the base common inerior of the sphere x² + y² + z² =80 and about the paraboloid z = 1/2 (x² + y² ).Volume = ∭dv From the equation of the sphere,x² + y² + z² = 80 .....(1)From the equation of the paraboloid, z = 1/2 (x² + y²) => x² + y² = 2z... (2)The projection of the intersection of the sphere and the paraboloid onto the xy-plane is the circle x² + y² = 80/3.The limits of integration for z are 0 and 80 - x² - y². Thus, the integral becomesV = ∬R(80 - x² - y²) dA where R is the region in the xy-plane bounded by the circle x² + y² = 80/3 (projection of the intersection of the sphere and the paraboloid).Converting to polar coordinates, we have x = rcosθ, y = rsinθ, and dA = r dr dθ. R is the circle x² + y² = 80/3, so the limits of integration for r are 0 and sqrt(80/3).Thus,V = ∫₀²π ∫₀sqrt(80/3) (80 - r²) r dr dθV = π/3 (6400/3 - 3200/3)sqrt(80/3) = (6400/9)πsqrt(80/3) Therefore, the integral to calculate the volume of the solid is:V = (6400/9)πsqrt(80/3)The graph of the solid
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5. Let H be the hemisphere H = {(x,y,z) € R³ : x² + y² + z² = 16, z ≤ 0} and F(x,y,z) = (0, 2y, -4). Compute the flux integral JF.Nas where N is directed in the direction positive z-coordinate
To compute the flux integral JF.Nas, where N is directed in the positive z-coordinate direction, we need to evaluate the surface integral over the hemisphere H with the vector field F(x, y, z) = (0, 2y, -4).
The surface integral can be computed using the formula JF.Nas = ∬ F · N dS, where F is the vector field, N is the unit normal vector to the surface, and dS represents the infinitesimal area element on the surface.
Since N is directed in the positive z-coordinate direction, it is given by N = (0, 0, 1).
To evaluate the surface integral, we need to parameterize the hemisphere H. We can use spherical coordinates to parameterize the surface, where x = r sinθ cosϕ, y = r sinθ sinϕ, and z = r cosθ, with the constraint r = 4 and θ ∈ [0, π/2] and ϕ ∈ [0, 2π].
Substituting the parameterization into F · N, we have F · N = (0, 2y, -4) · (0, 0, 1) = -4.
The surface integral becomes JF.Nas = ∬ -4 dS.
Integrating over the surface of the hemisphere H, we obtain the flux integral.
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The Edison Electric Institute has published figures on the number of kilowatt hours used annually by various home appliances. It is claimed that a vacuum cleaner uses an average of = 25 kilowatt hours per year. If a random sample of 10 homes included in a planned study indicates that vacuum cleaners use an average of 22 kilowatt hours per year with a standard deviation of 5.5 kilowatt hours, does this suggest at the 0.05 level of significance that vacuum cleaners use, on average is less than 25 kilowatt hours annually?
To determine whether vacuum cleaners use, on average, less than 25 kilowatt hours annually, a hypothesis test is conducted at the 0.05 level of significance. A random sample of 10 homes indicates an average usage of 22 kilowatt hours with a standard deviation of 5.5 kilowatt hours. The goal is to determine if this sample provides enough evidence to reject the null hypothesis that the average usage is equal to 25 kilowatt hours.
To conduct the hypothesis test, the null hypothesis (H0) is that the average usage of vacuum cleaners is 25 kilowatt hours annually, while the alternative hypothesis (Ha) is that the average usage is less than 25 kilowatt hours annually.
Next, the test statistic is calculated using the sample mean, population mean, sample standard deviation, and sample size. In this case, the sample mean is 22 kilowatt hours, the population mean (under the null hypothesis) is 25 kilowatt hours, the sample standard deviation is 5.5 kilowatt hours, and the sample size is 10.
The test statistic is then compared to the critical value from the t-distribution at the specified level of significance (0.05). If the test statistic is less than the critical value, the null hypothesis is rejected, indicating evidence in favor of the alternative hypothesis.
Using statistical software or a t-table, the test statistic is calculated and compared to the critical value. If the test statistic falls in the rejection region (i.e., is less than the critical value), it suggests that vacuum cleaners use, on average, less than 25 kilowatt hours annually, providing evidence to support the claim at the 0.05 level of significance.
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(2x³ + 6x²-7x -4)-(2x² + 9x - 3)
Answer:
2x³ + 4x² - 16x - 1
Step-by-step explanation:
this is the simplified answer. I hope this is what you were asking for.
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