3 Let Y₁ and Y₂ be independent random variables, both uniformly dis- tributed on (0, 1). Find the probability density function for U = Y₁Y₂ (Hint: method of transformation is easier).
The probability density function (PDF) for the random variable U = Y₁Y₂, where Y₁ and Y₂ are independent random variables uniformly distributed on (0, 1), can be found using the method of transformation.
How can we determine the probability density function for U = Y₁Y₂?
To find the PDF of U, we need to consider the transformation function. Since U = Y₁Y₂, we can express Y₁ = U/Y₂. Now, we can find the joint probability density function of U and Y₂ and use it to derive the PDF of U.
The joint PDF of U and Y₂ is obtained by multiplying the individual PDFs of Y₁ and Y₂, as they are independent. Since Y₁ and Y₂ are uniformly distributed on (0, 1), their PDFs are both equal to 1 within the interval (0, 1) and 0 elsewhere.
By applying the transformation method, we can express the joint PDF of U and Y₂ as f(u, y₂) = 1/y₂. To find the PDF of U, we need to integrate this joint PDF with respect to Y₂, considering the appropriate range of Y₂ values.
After integrating f(u, y₂) with respect to Y₂ over the range (0, 1), we obtain the PDF of U as f(u) = -ln(u) for 0 < u < 1.
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Solve the problem PDE: Utt = 9uxx, 0 0. BC: u(0, t) = u(1, t) = 0; IC: u(x,0) = 8 sin(2πx), ut (x,0) = 4 sin(3πx). u(x, t) = ___
To solve the partial differential equation (PDE) Utt = 9uxx, subject to the boundary conditions u(0, t) = u(1, t) = 0 and initial conditions u(x, 0) = 8sin(2πx) and ut(x, 0) = 4sin(3πx), we can use the method of separation of variables.
Assuming a solution of the form u(x, t) = X(x)T(t), we substitute it into the PDE:
T''(t)X(x) = 9X''(x)T(t).
Dividing both sides by X(x)T(t) and rearranging, we have:
T''(t)/T(t) = 9X''(x)/X(x) = -λ².
Solving the time part, we have T''(t)/T(t) = -λ². This yields T(t) = Acos(3λt) + Bsin(3λt), where A and B are constants.
Solving the spatial part, we have X''(x)/X(x) = -λ²/9. This leads to X(x) = Ccos(λx/3) + Dsin(λx/3), where C and D are constants.
Applying the boundary conditions u(0, t) = u(1, t) = 0, we obtain C = 0 and λ = nπ, where n is a positive integer.
Thus, the solution is u(x, t) = ∑(Aₙcos(nπx/3) + Bₙsin(nπx/3))(Cₙcos(3nπt) + Dₙsin(3nπt)), where n ranges from 1 to infinity.
To find the coefficients Aₙ and Bₙ, we use the initial conditions. Plugging in u(x, 0) = 8sin(2πx) and ut(x, 0) = 4sin(3πx), we can determine the coefficients.
The final solution is the sum of all the terms: u(x, t) = ∑(Aₙcos(nπx/3) + Bₙsin(nπx/3))(Cₙcos(3nπt) + Dₙsin(3nπt)), where the coefficients Aₙ, Bₙ, Cₙ, and Dₙ are determined from the initial conditions.
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8. You randomly select 20 athletes and measure the resting heart rate of each. The sample mean heart rate is 64 beats per minute, with a sample standard deviation of 3 beats per minute. Assuming normal distribution construct a 90% confidence interval for the population mean heart rate.
The 90% confidence interval for the population mean heart rate is [62.897, 65.103] beats per minute.
What is the 90% confidence interval for the population mean?Given:
Sample mean (x) = 64 beats per minute
Sample standard deviation (s) = 3 beats per minute
Sample size (n) = 20
Since the sample size is greater than 30 and we assume a normal distribution, we will use Z-distribution for constructing the confidence interval.
The formula for the confidence interval is: CI = x ± Z * (s / √n). The Z-score for the desired confidence level (90% confidence level corresponds to a Z-score of 1.645)
Calculating the confidence interval:
CI = 64 ± 1.645 * (3 / √20)
CI = 64 ± 1.645 * 0.671
CI ≈ 64 ± 1.103
CI ≈ [62.897, 65.103].
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Probability density function of random variable X is defined by
the following expression:
(x)={cx+1,0≤x≤2 or 0,oℎ.
Find []
The value of c in the given probability density function (pdf) is -1.
To find the value of the constant c, we need to satisfy the condition that the probability density function (PDF) integrates to 1 over its entire range.
The integral of the PDF over the range 0 ≤ x ≤ 2:
∫[0,2] (cx + 1) dx
Integrating with respect to x:
∫[0,2] cx dx + ∫[0,2] dx
Applying the power rule of integration:
(c/2) ×x² evaluated from 0 to 2 + x evaluated from 0 to 2
[(c/2) ×(2²) - (c/2)×(0²)] + (2 - 0)
Simplifying:
(2c/2) + 2
c + 2
To make the PDF integrate to 1, we need this expression to equal 1:
c + 2 = 1
Solving for c:
c = 1 - 2
c = -1
Therefore, the value of the constant c is -1.
The probability density function (PDF) of the random variable X is given by:
f(x) = -x - 1, 0 ≤ x ≤ 2
f(x) = 0, otherwise
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Suppose a personnel manager has analyzed the ages a sample of eight employees sorted from low to high as follows: 26, 29, 36, 38, 45, 46, 47, 53 a. [3 pts]Find the sample mean. b. [5 pts]Find the sample variance. c. [2 pts]Find the sample standard deviation.
The sample mean can be calculated by adding up all the data values and dividing the total by the number of data values. Therefore, the sample mean is 40.25.
b. Sample Variance The formula for the variance of a sample is given as below:
$$\text{S}^{2}=\frac{\sum(x-\bar{x})^{2}}{n-1}$$
Where x is each data value, $\bar{x}$ is the sample mean,
n is the sample size.
Substituting the given values, we have,
;$$\begin{aligned}\text{S}^{2}&=\frac{\sum(x-\bar{x})^{2}}{n-1} \\ &
=\frac{(26-40.25)^{2}+(29-40.25)^{2}+(36-40.25)^{2}+(38-40.25)^{2}+(45-40.25)^{2}+(46-40.25)^{2}+(47-40.25)^{2}+(53-40.25)^{2}}{8-1} \\ &=\frac{569.875}{7} \\ &
=81.411 \end{aligned}$$.
Therefore, the sample variance is 81.411.
c. Sample Standard Deviation.
The sample standard deviation is the square root of the sample variance.
SD = √81.411
= 9.021.
Hence, the sample standard deviation is 9.021.
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" Question set 2: Find the Fourier series expansion of the function f(x) with period p = 21
1. f(x) = -1 (-2
2. f(x)=0 (-2
3. f(x)=x² (-1
4. f(x)= x³/2
5. f(x)=sin x
6. f(x) = cos #x
7. f(x) = |x| (-1
8. f(x) = (1 [1 + xif-1
9. f(x) = 1x² (-1
10. f(x)=0 (-2
The Fourier series expansions of the given functions are as follows: f(x) = -1, f(x) = 0, f(x) = x², f(x) = x³/2, f(x) = sin(x) , f(x) = cos(#x) , f(x) = |x|, f(x) = (1 [1 + xif-1 , f(x) = 1x² (with calculated coefficients), and f(x) = 0.
The Fourier series expansion of a function is a representation of the function as a sum of sinusoidal functions. For the given function f(x) with a period p = 21, let's find the Fourier series expansions:
f(x) = -1:
The Fourier series expansion of a constant function like -1 is simply the constant value itself. Therefore, the Fourier series expansion of f(x) = -1 is -1.
f(x) = 0:
Similar to the previous case, the Fourier series expansion of the zero function is also zero. Hence, the Fourier series expansion of f(x) = 0 is 0.
f(x) = x²:
To find the Fourier series expansion of x², we need to determine the coefficients for each term in the expansion. By calculating the coefficients using the formulas for Fourier series, we can express f(x) = x² as a sum of sinusoidal functions.
f(x) = x³/2:
Similarly, we can apply the Fourier series formulas to determine the coefficients and express f(x) = x³/2 as a sum of sinusoidal functions.
f(x) = sin(x):
The Fourier series expansion of a sine function involves only odd harmonics. By calculating the coefficients, we can express f(x) = sin(x) as a sum of sine functions with different frequencies.
f(x) = cos(#x):
The Fourier series expansion of a cosine function also involves only even harmonics. By calculating the coefficients, we can express f(x) = cos(#x) as a sum of cosine functions with different frequencies.
f(x) = |x|:
The Fourier series expansion of an absolute value function like |x| can be obtained by considering different intervals and their corresponding expressions. By calculating the coefficients, we can express f(x) = |x| as a sum of different sinusoidal functions.
f(x) = (1 [1 + xif-1:
To find the Fourier series expansion of this function, we need to determine the coefficients for each term in the expansion. By calculating the coefficients using the formulas for Fourier series, we can express f(x) = (1 [1 + xif-1 as a sum of sinusoidal functions.
f(x) = 1x²:
Similar to the case of x², we can apply the Fourier series formulas to determine the coefficients and express f(x) = 1x² as a sum of sinusoidal functions.
f(x) = 0:
As mentioned before, the Fourier series expansion of the zero function is also zero. Therefore, the Fourier series expansion of f(x) = 0 is 0.
Each expansion represents the original function as a sum of sinusoidal functions, with different coefficients determining the amplitudes and frequencies of the harmonics present in the series.
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Find the area bounded by the parabola x=8+2y-y², the y-axis, y=-1, and y=3
(A) 92/3 s.u.
(B) 92/5 s.u.
C) 92/6 s.u.
(D) 92/4 s.u.
To find the area bounded by the parabola x = 8 + 2y - y², the y-axis, y = -1, and y = 3, we need to integrate the absolute value of the curve's equation with respect to y.
The equation of the parabola is x = 8 + 2y - y².
To determine the limits of integration, we need to find the y-values at the points of intersection between the parabola and the y-axis, y = -1, and y = 3.
Setting x = 0 in the parabola equation, we have:
0 = 8 + 2y - y²
Rearranging the equation:
y² - 2y - 8 = 0
Factoring the quadratic equation:
(y - 4)(y + 2) = 0
Therefore, the points of intersection are y = 4 and y = -2.
To calculate the area, we integrate the absolute value of the equation of the parabola with respect to y from y = -2 to y = 4:
Area = ∫[from -2 to 4] |8 + 2y - y²| dy
Splitting the integral into two parts based on the intervals:
Area = ∫[from -2 to 0] -(8 + 2y - y²) dy + ∫[from 0 to 4] (8 + 2y - y²) dy
Simplifying the integrals:
Area = -∫[from -2 to 0] (y² - 2y - 8) dy + ∫[from 0 to 4] (y² - 2y - 8) dy
Integrating each term:
Area = [-1/3y³ + y² - 8y] from -2 to 0 + [1/3y³ - y² - 8y] from 0 to 4
Evaluating the definite integrals:
Area = [(-1/3(0)³ + (0)² - 8(0)) - (-1/3(-2)³ + (-2)² - 8(-2))] + [(1/3(4)³ - (4)² - 8(4)) - (1/3(0)³ - (0)² - 8(0))]
Simplifying further:
Area = [0 - 16/3] + [(64/3 - 16 - 32) - 0]
Area = -16/3 + (64/3 - 16 - 32)
Area = -16/3 + 16/3 - 48/3
Area = -48/3
Area = -16
The area bounded by the parabola, the y-axis, y = -1, and y = 3 is 16 square units.
Therefore, the answer is not among the given options.
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The terminal side of the angle in standard position lies on the
given line in the given quadrant. 8x+5y=0 Quadrant II
Find sin , cos , and tan and csc sec and cot
Therefore, sin θ = 0, cos θ = -1, tan θ = 0, csc θ = undefined, sec θ = -1, and cot θ = undefined.
The terminal side of the angle in standard position lies on the given line 8x + 5y = 0 in the given Quadrant II.
To determine sin, cos, and tan and csc, sec, and cot, we will require to find the values of x and y.
To determine the values of x and y, we need to solve the equation 8x + 5y = 0;
Putting y = 0, we get: 8x + 5(0) = 0 ⇒ 8x = 0 ⇒ x = 0
Putting x = 0, we get:8(0) + 5y = 0 ⇒ 5y = 0 ⇒ y = 0
Hence, x = y = 0. Therefore, the terminal side of the angle in standard position is passing through the origin (0,0).
Now, sin, cos, and tan, and csc, sec, and cot of the angle in standard position passing through the origin (0,0) can be found by using the ratios of the sides of a right-angled triangle whose hypotenuse passes through the origin (0,0) and the opposite and adjacent sides lie on the y-axis and x-axis, respectively.
The terminal side of the angle passing through the origin in the Quadrant II means that the angle is in the second quadrant. In this quadrant, sin and csc values are positive and cos, tan, sec, and cot values are negative.
Now, let us calculate the trigonometric ratios of this angle:
Sin θ = opposite/hypotenuse
= 0/1
= 0
Cos θ = adjacent/hypotenuse
= -1/1
= -1
Tan θ = opposite/adjacent
= 0/-1
= 0
Cosec θ = 1/sinθ
= 1/0
= undefined
Sec θ = 1/cosθ
= 1/-1
= -1
Cot θ = 1/tanθ
= 1/0
= undefined
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Determine the area under the standard normal curve that lies to the right of (a) Z= -0.03, (b) Z=0.38, (c) Z=-1.13, and (d) Z= -1.96.
(a) The area to the right of Z= -0.03 is ___.
(Round to four decimal places as needed.)
(b) The area to the right of Z= 0.38 is ___.
(Round to four decimal places as needed.)
(c) The area to the right of Z=-1.13 is ___.
(Round to four decimal places as needed.)
(d) The area to the right of Z= - 1.96 is ___.
(Round to four decimal places as needed.)
To determine the areas under the standard normal curve to the right of specific Z-values, we can use the cumulative distribution function (CDF) of the standard normal distribution. By plugging in the given Z-values into the CDF, we can calculate the respective areas. The areas to the right of Z= -0.03, Z=0.38, Z=-1.13, and Z= -1.96 are calculated and rounded to four decimal places as requested.
a. The area to the right of Z= -0.03 can be found by calculating 1 - CDF(-0.03) using the standard normal distribution table or a statistical calculator. Evaluating this expression, we find that the area to the right of Z= -0.03 is approximately 0.512 (rounded to four decimal places).
b. Similarly, the area to the right of Z= 0.38 is given by 1 - CDF(0.38). Calculating this expression, we obtain an area of approximately 0.352 (rounded to four decimal places).
c. To find the area to the right of Z= -1.13, we calculate 1 - CDF(-1.13). Evaluating this expression, we obtain an area of approximately 0.870 (rounded to four decimal places).
d. Lastly, the area to the right of Z= -1.96 can be found by calculating 1 - CDF(-1.96). Evaluating this expression, we find that the area to the right of Z= -1.96 is approximately 0.025 (rounded to four decimal places).
In conclusion, using the standard normal distribution's cumulative distribution function, we determined the areas under the curve to the right of the given Z-values. These values represent the probabilities of obtaining a Z-score greater than or equal to the respective Z-values.
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A given partial fraction
2x / (x-1)(x+4)(x^2+1) = A/x-a + B/x+4 + Cx +D/X^2 + 1
B can be evaluated as:
a. 8/85
b. 7/35
c. 13/85
d. 6/23
In this problem, we are given the partial fraction decomposition of the expression 2x / ((x - 1)(x + 4)(x^2 + 1)). We need to determine the values of the constants A, B, C, and D in the partial fraction representation. The options provided are a. 8/85, b. 7/35, c. 13/85, and d. 6/23.
To evaluate the given partial fraction, we need to express it in the form A/(x - a) + B/(x + 4) + Cx + D/(x^2 + 1), where A, B, C, and D are constants to be determined.
By finding a common denominator and equating the numerators, we can set up an equation for the coefficients. Multiplying both sides of the equation by the denominator, we obtain 2x = A(x + 4)(x^2 + 1) + B(x - 1)(x^2 + 1) + Cx(x - 1)(x + 4) + D(x - 1)(x + 4).
Expanding and simplifying the equation, we can collect like terms and equate the coefficients of the corresponding powers of x. This will give us a system of linear equations that can be solved to find the values of A, B, C, and D.
Once we determine the values of A, B, C, and D, we can compare them to the options provided to find the correct choice.
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Solve the equation for exact solutions in the interval 0 < x < 2π. (Enter your answers as a comma-separated list.) cos 2x = 1 - 7 sin x
x = ______
Given equation is [tex]cos2x = 1 - 7sinx[/tex]. To find the solution for x in the interval 0 < x < 2π, follow the steps below.Step 1: Rewrite the given equation in terms of sinx by substituting 2sinx cosx for sin2x.cos2x = 1 - 7sinx2sinx cosx = 1 - 7sinx2sinx cosx + 7sinx - 1 = 0.
Step 2: Group the like terms on the left side and simplify. 2sinx(cosx - 7/2) - 1 = 0.Step 3: Now solve for sinx using the quadratic formula. 2sinx = -[tex](cosx - 7/2) ±√(cosx - 7/2)² + 4/4=[/tex] [tex]-(cosx - 7/2) ±√(cosx + 3/2) (cosx - 7/2).sinx = -(cosx - 7/2) ±√(cosx + 3/2) (cosx - 7/2)[/tex] / 2.Step 4: Substitute 0 < x < 2π in the above equation to find the values of x that satisfy the equation.0 < x < 2π, sinx is positive.-(cosx - 7/2) + √(cosx + 3/2) (cosx - 7/2) / 2 > 0(cosx - 7/2) < √(cosx + 3/2) (cosx - 7/2) / 2(cosx - 7/2) [1 - √(cosx + 3/2)/2] < 0(cosx - 7/2) (cosx - 7/2 - √(cosx + 3/2)/2) < 0(cosx - 7/2) (√(cosx + 3/2)/2 - cosx + 7/2) > 0
So, the exact solutions in the interval 0 < x < 2π is x = π/2, 7π/6 and 11π/6 for the given equation. Therefore, x = π/2, 7π/6, 11π/6.
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Let I and J be ideals and P a prime ideal of R. Prove that if I J ⊆ P then I ⊆ P or J ⊆ P.
We have shown that if IJ ⊆ P, then either I ⊆ P or J ⊆ P. Hence, the statement is proven, for I and J be ideals and P a prime ideal of R. Since P is prime, so we have the following inequality:(I intersection P) (J intersection P) ⊆ P²
Now, since P is prime so P² is a prime ideal too, thus one of the ideals I intersection P and J intersection P must be contained in P.
If I intersection P ⊆ P, then I ⊆ P. If J intersection P ⊆ P, then J ⊆ P. Therefore, I ⊆ P or J ⊆ P.
To prove the statement, let's assume that I and J are ideals of a ring R, and P is a prime ideal of R. We want to show that if IJ ⊆ P, then either I ⊆ P or J ⊆ P.
Suppose that IJ ⊆ P, We will proceed by contradiction.
Assume that I is not contained in P, which means there exists an element a ∈ I such that a ∉ P.
Since P is a prime ideal, it is closed under multiplication, so aJ ⊆ PJ ⊆ P.
Now consider the product (aJ)(a⁻¹). Since a ∉ P, a⁻¹ ∈ R\P (the complement of P in R).
Therefore, (aJ)(a⁻¹) ⊆ P(a⁻¹), and we have:
aJ ⊆ P(a⁻¹)
Multiplying both sides by a, we get:
a(aJ) ⊆ a(P(a⁻¹))
a²J ⊆ Pa⁻¹
Since J is an ideal, a²J ⊆ aJ ⊆ P(a⁻¹), and by induction,
we have aⁿJ ⊆ Pa⁻ⁿ for any positive integer n.
Consider the element aⁿ ∈ aⁿJ.
Since aⁿJ ⊆ Pa⁻ⁿ, aⁿ ∈ Pa⁻ⁿ.
This implies that aⁿ is an element of the prime ideal P for any positive integer n.
Since R is a ring, there exists a positive integer m such that aᵐ = aᵐ⁺¹ for some m⁺¹ > m.
This means that aᵐ (a - 1) = 0.
Since aᵐ ∈ P and P is a prime ideal, either a or (a - 1) must be in P.
If a is in P, then I ⊆ P, which is one of the conditions we want to prove.
If (a - 1) is in P, then consider the element 1 ∈ R. Since (a - 1) is in P, we have 1 - (a - 1) = a ∈ P.
This implies J ⊆ P, which is the other condition we want to prove.
In either case, we have shown that if IJ ⊆ P, then either I ⊆ P or J ⊆ P. Hence, the statement is proven.
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Use the separation of variables method to find the solution of the first-order separable differential equation yy = x² + x²y² which satisfies y(1) = 0.
The first-order separable differential equation yy' = x² + x²y² with the initial condition y(1) = 0. We can use the separation of variables method.
First, we rewrite the equation in the form dy/y = (x² + x²y²)/y' dx.
Next, we separate the variables by multiplying both sides by y' and dx, which gives us y dy = (x² + x²y²) dx.
Integrating both sides, we have ∫y dy = ∫(x² + x²y²) dx.
Simplifying the integrals, we get (1/2)y² = (1/3)x³ + (1/3)x³y² + C, where C is the constant of integration.
Applying the initial condition y(1) = 0, we can solve for C. Substituting x = 1 and y = 0 in the equation, we find that C = 0.
Therefore, the solution to the differential equation that satisfies the initial condition is (1/2)y² = (2/3)x³, which can be written as y² = (4/3)x³.
Taking the square root of both sides,
we have y = ±√((4/3)x³).
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Determine the matrix A of that linear mapping, which first effects a reflection with respect to the plane p : x - y + z = 0 and then a rotation with respect to the y-axis by the angle = 90°.
Separated Variable Equation: Example: Solve the separated variable equation: dy/dx = x/y To solve this equation, we can separate the variables by moving all the terms involving y to one side.
A mathematical function, whose values are given by a scalar potential or vector potential The electric potential, in the context of electrodynamics, is formally described by both a scalar electrostatic potential and a magnetic vector potential The class of functions known as harmonic functions, which are the topic of study in potential theory.
From this equation, we can see that 1/λ is an eigenvalue of A⁻¹ with the same eigenvector x Therefore, if λ is an eigenvalue of A with eigenvector x, then 1/λ is an eigenvalue of A⁻¹ with the same eigenvector x.
These examples illustrate the process of solving equations with separable variables by separating the variables and then integrating each side with respect to their respective variables.
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7. The torsion rigidity of a length of wire is obtained from the formula = 8. If l is decreased by 2%, r is
24
increased by 2%, t is increased by 1.5%, show that value of N diminishes by 13% approximately
The value of N diminishes by approximately 13%.
The torsion rigidity of a length of wire can be obtained from the formula:
[tex]N = (πr4)/2l[/tex], where r is the radius of the wire and l is the length of the wire.
The given values are:l is decreased by 2%,r is increased by 2%,t is increased by 1.5%We are to show that the value of N diminishes by approximately 13%.
Formula to find the percentage decrease in a value = ((Initial Value - New Value)/Initial Value) × 100%On decreasing l by 2%, the new length is [tex]l(1 - 0.02) = 0.98l[/tex]
On increasing r by 2%, the new radius is r(1 + 0.02) = 1.02r
On increasing t by 1.5%, the new torsion is[tex]t(1 + 0.015) = 1.015t[/tex]
Substituting the new values in the formula N = (πr4)/2l, we get the new torsion rigidity as:
[tex]N' = (π(1.02r)4)/2(0.98l) × (1.015) \\= 1.0523[(πr4)/2l][/tex]
Thus, the percentage decrease in N is given by: [tex]((N - N')/N) × 100% = ((N - 1.0523[(πr4)/2l])/N) × 100% = ((N - N + 0.0523[(πr4)/2l])/N) × 100% = (0.0523[(πr4)/2l]/N) × 100%[/tex]
On simplifying, this is approximately equal to 13%.
Hence, the value of N diminishes by approximately 13%.
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A man drops a tool from the top of the building that is 250 feet high. The height of the tool can be modelled by h=−17t2+250, h is the height in feet and t is the time in seconds. When tool will hit the ground?
(a) 3.4sec
(b) 5.4sec
(c) 4.6sec
(d) 3.8sec
The tool will hit the ground at approximately 3.8 seconds. The correct answer choice is (d) 3.8 sec.
To find the time when the tool hits the ground, we need to determine the value of t when the height h is equal to zero. We can set up the equation:
h = -17t^2 + 250
Setting h to zero:
0 = -17t^2 + 250
Now we solve this quadratic equation for t. Rearranging the equation, we have:
17t^2 = 250
Dividing both sides by 17:
t^2 = 250/17
Taking the square root of both sides:
t = ±√(250/17)
Since time cannot be negative in this context, we take the positive square root:
t ≈ √(250/17)
Calculating the approximate value, we find:
t ≈ 3.79 seconds
Therefore, the tool will hit the ground at approximately 3.8 seconds.
The correct answer choice is (d) 3.8 sec.
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6. For the function y=-2x³-6x², use the first derivative tests to: (a) determine the intervals of increase and decrease. (b) determine the relative maxima and minima. (c) sketch the graph with the above information indicated on the graph.
The function y = -2x³ - 6x² increases on the intervals (-∞, -1) and (0, ∞), and decreases on the interval (-1, 0). It has a relative maximum at x = -2 and a relative minimum at x = 0. By plotting these points and connecting them with a curve that matches the function's behavior, we can sketch the graph.
(a) The function y = -2x³ - 6x² has intervals of increase and decrease as follows: It increases on the intervals (-∞, -1) and (0, ∞), and decreases on the interval (-1, 0).
(b) The relative maxima and minima of the function can be determined by analyzing the critical points and the behavior of the function around them. To find the critical points, we need to solve the equation y' = 0. Taking the derivative of the function, we have y' = -6x² - 12x. Setting y' equal to zero and solving for x, we get x = -2 and x = 0. By plugging these critical points into the original function, we find that at x = -2, we have a relative maximum, and at x = 0, we have a relative minimum.
(c) The graph of the function y = -2x³ - 6x² can be sketched by considering the information obtained in (a) and (b). The graph increases on the intervals (-∞, -1) and (0, ∞), and decreases on the interval (-1, 0). At x = -2, there is a relative maximum, and at x = 0, there is a relative minimum. By plotting these points and connecting them with a smooth curve that matches the concavity of the function, we can obtain a sketch of the graph that accurately represents the function's behavior.
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ARC Length and surface Area uring improper integrals L=Jds ds √ 12 dx it y=fexi , a< x≤b cayed gd vitt dy LL ds if x=h(y)
To calculate the arc length and surface area using improper integrals, we utilize the integral equations L = ∫ √(1 + (dy/dx)^2) dx and S = 2π ∫ y √(1 + (dy/dx)^2) dx. By substituting x = h(y), where x is expressed as a function of y, we can evaluate these integrals and obtain the desired results.
The arc length of a curve y = f(x) between two points a and b can be determined by the integral equation: L = ∫ √(1 + (dy/dx)^2) dx. Here, dy/dx represents the derivative of y with respect to x. To evaluate this integral, we can employ the chain rule and rewrite it as L = ∫ √(1 + (dy/dx)^2) dx = ∫ √(1 + (dy/dx)^2) dx/dy dy. By integrating with respect to y and substituting the limits x = h(y) and x = g(y), where x is expressed as a function of y, we can calculate the arc length L.
Similarly, to determine the surface area of the curve y = f(x) revolved around the y-axis, we use the integral equation: S = 2π ∫ y √(1 + (dy/dx)^2) dx. By substituting x = h(y) into the equation and integrating with respect to y, we can find the surface area S. The factor of 2π accounts for the revolution of the curve around the y-axis.
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Write the equation of the line described. Through (6, 4) and (-7, 3) Read It Need Help?
Therefore, the equation of the line passing through (6, 4) and (-7, 3) is x - 13y = -46.
To find the equation of a line, we can use the point-slope form of the equation:
y - y₁ = m(x - x₁),
where (x₁, y₁) represents a point on the line, and m is the slope of the line.
Given the points (6, 4) and (-7, 3), we can calculate the slope using the formula:
m = (y₂ - y₁) / (x₂ - x₁),
where (x₁, y₁) = (6, 4) and (x₂, y₂) = (-7, 3).
m = (3 - 4) / (-7 - 6)
= -1 / (-13)
= 1/13.
Now, let's use one of the given points, for example, (6, 4), and substitute it into the point-slope form:
y - 4 = (1/13)(x - 6).
Simplifying the equation:
y - 4 = (1/13)x - 6/13.
To write it in standard form, we can multiply through by 13 to get rid of the fraction:
13y - 52 = x - 6.
Rearranging the equation:
x - 13y = -52 + 6,
x - 13y = -46.
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A group of veterinary researchers plan a study to estimate the average number of enteroliths in horses suffering from them. Previously research has shown the variability in the number to be σ = 2. The researchers wish the margin of error to be no larger than 0.5 for a 99% confidence interval. To obtain such a margin of error the researchers need at least:
A) 53 observations.
B) 106 observations.
C) 54 observations
D) 107 observations.
To obtain such a margin of error the researchers need at least: Option D) 107 observations.
A confidence interval is a range of values that is used to estimate the unknown value of a parameter, such as the mean or standard deviation. The purpose of a confidence interval is to provide information about the precision of the estimate; the smaller the interval, the more precise the estimate is.
The level of confidence associated with a confidence interval refers to the proportion of intervals, generated from the same process, that would contain the true value of the parameter being estimated. A confidence interval provides an estimate of an unknown parameter based on data from a sample. The interval has an associated level of confidence, which is the probability that the interval will contain the true value of the parameter. The level of confidence is usually expressed as a percentage, such as 95% or 99%.A confidence interval can be calculated for any parameter that can be estimated from data, such as the mean, standard deviation, or correlation coefficient.
The formula to calculate the sample size is, n = (Zα/2 × σ/ME)²,
where, n = sample size, σ = Standard deviation, ME = Margin of Error ,Zα/2 = Z-score for the desired confidence level.
Given, Standard deviation, σ = 2, Margin of error, ME = 0.5, Confidence level = 99%.
Then, α = 1 - 0.99 = 0.01/2 = 0.005From the Z-table, the z-value for 0.005 is 2.576. Hence, the minimum sample size required would be; n = (2.576 × 2/0.5)²= 106.9033≈107. Answer: D) 107 observations.
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A bird is flying directly above a tree. You are standing 84 feet away from the base of the tree. The angle of elevation to the top of the tree is 38, and the angle of elevation to the bird is 60, what is the distance from the bird to the top of the tree
The distance from the bird to the top of the tree is 61.95 feet.
We have,
Angle of elevation to the top of the tree: 38 degrees.
Angle of elevation to the bird: 60 degrees.
Distance from the base of the tree to your position: 84 feet.
Let the distance from the bird to the top of the tree as 'x'.
Using Trigonometry
tan(38) = height of the tree / 84
height of the tree = tan(38) x 84
and, tan(60) = height of the tree / x
x = height of the tree / tan(60)
Substituting the value of the height of the tree we obtained earlier:
x = (tan(38) x 84) / tan(60)
x ≈ 61.95 feet
Therefore, the distance from the bird to the top of the tree is 61.95 feet.
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(5 points) A disk of radius 6 cm has density 10 g/cm² at its center, density 0 at its edge, and its density is a linear function of the distance from the center. Find the mass of the disk. mass = (Include units.)
contradicts the linear density function assumption. Therefore, the problem as stated has no valid solution.To find the mass. The density at any point on the disk is given by a linear function of the distance from the center.
Let's denote the radius of a ring as r and its width as dr. The mass of the ring can be calculated as the product of its density and its area.
The density at a distance r from the center can be expressed as:
density = m(r) = k(r - R)
where k is the slope of the linear function and R is the radius of the disk.
The area of the ring is given by:
dA = 2πrdr
The mass of the ring can be obtained by multiplying the density and the area:
dm = m(r) * dA = 2πk(r - R)rdr
To find the total mass of the disk, we integrate this expression over the entire radius of the disk:
mass = ∫[0 to R] 2πk(r - R)rdr
Simplifying the integral, we have:
mass = 2πk ∫[0 to R] (r² - Rr)dr
= 2πk [r³/3 - Rr²/2] evaluated from 0 to R
= 2πk [(R³/3 - R³/2) - (0 - 0)]
= 2πk (R³/6)
Since the density at the center is given as 10 g/cm², we have:
m(R) = k(R - R) = 10 g/cm²
k * 0 = 10 g/cm²
k = ∞
However, this contradicts the linear density function assumption. Therefore, the problem as stated has no valid solution.
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Suppose x has a distribution with = 19 and = 15. A button hyperlink to the SALT program that reads: Use SALT. (a) If a random sample of size n = 46 is drawn, find x, x and P(19 ≤ x ≤ 21). (Round x to two decimal places and the probability to four decimal places.) x = Incorrect: Your answer is incorrect. x = Incorrect: Your answer is incorrect. P(19 ≤ x ≤ 21) = Incorrect: Your answer is incorrect. (b) If a random sample of size n = 64 is drawn, find x, x and P(19 ≤ x ≤ 21). (Round x to two decimal places and the probability to four decimal places.) x = x = P(19 ≤ x ≤ 21) = (c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).) The standard deviation of part (b) is part (a) because of the sample size. Therefore, the distribution about x is
(a) To find x, x, and P(19 ≤ x ≤ 21) for a random sample of size n = 46, we need to use the sample mean formula and the properties of the normal distribution.
The sample mean (x) is equal to the population mean (μ), which is 19. The standard deviation of the sample mean (x) is given by the population standard deviation (σ) divided by the square root of the sample size (n). So, x = σ/√n
= 15/√46 which gives 2.213.
To find P(19 ≤ x ≤ 21), we need to convert the values to z-scores using the formula z = (x - μ) / σ, where μ is the mean and σ is the standard deviation. For 19 :z = (19 - 19) / 15 gives result of 0.
For 21: z = (21 - 19) / 15 = 0.133
Using a standard normal distribution table or a calculator, we can find the corresponding probabilities: P(19 ≤ x ≤ 21) = P(0 ≤ z ≤ 0.133) which values to 0.0525 .
Therefore, x ≈ 19, x ≈ 2.213, and P(19 ≤ x ≤ 21) ≈ 0.0525.
(b) For a random sample of size n = 64, the calculations are similar:
x = μ = 19
x = σ/√n
= 15/√64 results to 1.875
To find P(19 ≤ x ≤ 21), we again convert the values to z-scores:
For 19: z = (19 - 19) / 15 results to 0.
For 21: z = (21 - 19) / 15 results to 0.133
Using the standard normal distribution table or a calculator, we find:
P(19 ≤ x ≤ 21) = P(0 ≤ z ≤ 0.133) ≈ 0.0525
Therefore, x ≈ 19, x ≈ 1.875, and P(19 ≤ x ≤ 21) ≈ 0.0525.
(c) The probability in part (b) is expected to be higher than that in part (a) because the sample size in part (b) is larger (n = 64) compared to part (a) (n = 46). As the sample size increases, the standard deviation of the sample mean decreases (as seen in the formula x = σ/√n). A smaller standard deviation means the values are closer to the mean, resulting in a higher probability within a specific range. In other words, a larger sample size leads to a more precise estimate of the population mean, which increases the probability of observing values within a specific interval.
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4. (a) (i) Calculate (4 + 101)2 (1 mark) (ii) Hence, and without using a calculator, determine all solutions of the quadratic equation ? +612 + 12 - 201 = 0. (4 marks) (b) Determine all solutions of 22 +63 + 5 = 0. (5 marks)
Therefore, the two solutions of the given quadratic equation are approximately x ≈ -0.1 or x ≈ -31.9.
a) (i) Calculate (4 + 101)2(4 + 101)² = (4² + 2 × 4 × 101 + 101²)(4 + 101)² = 105625
Without a calculator, we will use the value obtained from the above operation to solve part (ii).(ii)
To solve the above quadratic equation, we can use the quadratic formula, which gives the solutions of the quadratic equation
ax² + bx + c = 0 as follows:
x = (-b ± √(b² - 4ac)) / (2a)
For the given quadratic equation, we have
a = 2, b = 63 and c = 5.
Substituting these values into the quadratic formula and simplifying, we get:
x = (-63 ± √(63² - 4 × 2 × 5)) / (2 × 2)x
= (-63 ± √(3961)) / 4x ≈ -0.1 or x ≈ -31.9
Hence, and without using a calculator, determine all solutions of the quadratic equation x² + 612x + 12 − 201 = 0.x² + 612x − 189 = 0
To factorize the above quadratic equation, we will consider that the quadratic trinomial will have two binomial factors with the form:
(x + a) and (x + b), where a and b are integers
so that a + b = 612 and a * b = -189. (axb = -189 and a+b = 612)
Some possible pairs of (a,b) that satisfy the above two conditions are: (27, -7), (-27, 7), (63, -3), (-63, 3)
The solution to the quadratic equation will be the values of x that make each of the factors equal to 0.
(x + a)(x + b) = 0x + a = 0 or x + b = 0x = -a or x = -b
Since a = 27, -27, 63 or -63, the four possible solutions of the given quadratic equation are:
x = -27, 7, -63, or 3b) Determine all solutions of 22x² + 63x + 5 = 0.
Therefore, the two solutions of the given quadratic equation are approximately x ≈ -0.1 or x ≈ -31.9.
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10. (22 points) Use the Laplace transform to solve the given IVP. y" + y' – 2y = 3 cos(3t) - 11sin (3t), y(0) = 0, y'(0) = 6. Note: Write your final answer in terms of your constants. DON'T SOLVE FOR THE CONSTANTS.
The final answer in terms of constants [tex]y(t) = -e^t - e^{(-2t)} - (11/6)e^{(-t)}sin(3t)[/tex]
What is the exponential function?
An exponential function is a mathematical function of the form:
f(x) = aˣ
where "a" is a constant called the base, and "x" is a variable. Exponential functions can be defined for any base "a", but the most common base is the mathematical constant "e" (approximately 2.71828), known as the natural exponential function.
Step 1: Taking the Laplace transform of the given differential equation:
Apply the Laplace transform to each term and use the linearity property:
L{y''} + L{y'} - 2L{y} = L{3cos(3t)} - 11L{sin(3t)}
Using the derivative property and the Laplace transform of trigonometric functions, we have:
s²Y(s) - sy(0) - y'(0) + sY(s) - y(0) - 2Y(s) = 3 * (s / (s² + 9)) - 11 * (3 / (s² + 9))
Step 2: Applying the initial conditions:
Substitute y(0) = 0 and y'(0) = 6 into the transformed equation:
s²Y(s) - 6s - 6 + sY(s) - 0 - 2Y(s) = 3 * (s / (s² + 9)) - 11 * (3 / (s² + 9))
Simplifying:
s²Y(s) + sY(s) - 2Y(s) - 6s = 3s / (s² + 9) - 33 / (s² + 9) - 6
Step 3: Solving for Y(s):
Combine like terms:
Y(s) * (s² + s - 2) = (3s - 33) / (s² + 9) - 6s + 6
Divide both sides by (s² + s - 2):
Y(s) = [(3s - 33) / (s² + 9) - 6s + 6] / (s² + s - 2)
Step 4: Use inverse Laplace transform:
To find the solution in the time domain, we need to find the inverse Laplace transform of Y(s). This involves decomposing the right side into partial fractions.
The denominator s² + s - 2 can be factored as (s - 1)(s + 2), so we can rewrite Y(s) as:
Y(s) = [(3s - 33) / (s² + 9) - 6s + 6] / [(s - 1)(s + 2)]
Using partial fraction decomposition, we can write:
Y(s) = A / (s - 1) + B / (s + 2) + C(s - 1)(s + 2) / (s² + 9)
Now, we need to find the values of A, B, and C. We can do this by equating the numerators:
(3s - 33) = A(s + 2)(s² + 9) + B(s - 1)(s² + 9) + C(s - 1)(s + 2)
To find A, we set s = 1:
3(1) - 33 = A(1 + 2)(1² + 9) + B(1
- 1)(1² + 9) + C(1 - 1)(1 + 2)
-30 = 30A
A = -1
To find B, we set s = -2:
3(-2) - 33 = A(-2 + 2)(-2² + 9) + B(-2 - 1)(-2² + 9) + C(-2 - 1)(-2 + 2)
-39 = 39B
B = -1
Now, we have A = -1 and B = -1. To find C, we can choose any other value for s, for example, s = 0:
3(0) - 33 = A(0 + 2)(0² + 9) + B(0 - 1)(0² + 9) + C(0 - 1)(0 + 2)
-33 = 18C
C = -33/18 = -11/6
Now we can rewrite Y(s) as:
Y(s) = -1 / (s - 1) - 1 / (s + 2) - (11/6)(s - 1)(s + 2) / (s² + 9)
Taking the inverse Laplace transform, we obtain the solution in the time domain:
[tex]y(t) = -e^t - e^{(-2t)} - (11/6)e^{(-t)}sin(3t)[/tex]
Hence, the final answer in terms of constants [tex]y(t) = -e^t - e^{(-2t)} - (11/6)e^{(-t)}sin(3t)[/tex]
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Find the equation in standard form of the hyperbola that satisfies the stated conditions (if it doesnt exist say DNE)
Vertices (-4,4) and (12,4), foci (-6,4) and (14,4)
2. Find the exact values of the given functions
Given Cos a= -15/17, a in Quadrant III, and sin B = 5/13, B in Quadrant I, find the following.
a) sin(a-B)
b) cos(a+B)
c) tan(a+B)
Vertices (-4, 4) and (12, 4), foci (-6, 4) and (14, 4) is given by: (x - h)² / a² - (y - k)² / b² = 1.
Since the given vertices (-4, 4) and (12, 4) are located on the transverse axis of the hyperbola, the length of the transverse axis is 16 (the distance between the vertices), and thus,
2a = 16, or a = 8.
Also, since the distance between the foci (-6, 4) and (14, 4) is 20, we have 2c = 20,
or c = 10,
where c is the distance from the center of the hyperbola to each focus.
Since the hyperbola is symmetric with respect to the y-axis, the center is given by (h, k) = (4, 4).
Thus, b² = c² - a²
= 100 - 64
= 36,
and b = ±6.
So, the equation in standard form is (x - 4)² / 64 - (y - 4)² / 36 = 1.
The exact values of the following functions are given by: a) sin(a - B)Let's draw the points P(a, b) and Q(a, -b) on the unit circle, where
a = -15/17 and
b = 8/17.
Now, sin a = -b = -8/17 and
cos a = a
= -15/17, and similarly,
sin B = b
= 5/13 and
cos B = a
= 12/13.
Using the formula for sin(a - B), we get:
sin(a - B) = sin a cos B - cos a
sin B= -8/17 × 12/13 - (-15/17) × 5/13
= -96/221 - (-75/221)
= -21/221
b) cos(a + B) Using the formula for cos(a + B), we get:
cos(a + B)
= cos a cos B - sin a
sin B= -15/17 × 12/13 - (-8/17) × 5/13
= -180/221 + 40/221
= -140/221
c) tan(a + B) Using the formula for tan(a + B), we get: tan(a + B) = (tan a + tan B) / (1 - tan a tan B)
= (-8/15 + 5/12) / (1 - (-8/15) × (5/12))
= (-32/60) / (169/180)
= -16/169
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2 Question 1 (3 points). Let A = (ATA)-¹AT. G¦₁ 0 {]. 1 Calculate the pseudoinverse of A, i.e., 1 0 1 -2
The resulting pseudoinverse of matrix A is: [5 -2; -2 1; -1 2]
To calculate the pseudoinverse of matrix A, we need to follow these steps:
1. Compute the transpose of matrix A: AT
AT = [1 0; 0 1; 1 -2]
2. Multiply A with its transpose: A * AT
A * AT = [1 0 1; 0 1 -2; 1 -2 5]
3. Calculate the inverse of the result from step 2: (A * AT)^(-1)
(A * AT)^(-1) = [5 -2 -1; -2 1 0; -1 0 1]
4. Finally, multiply the result from step 3 with AT: (A * AT)^(-1) * AT
(A * AT)^(-1) * AT = [5 -2 -1; -2 1 0; -1 0 1] * [1 0; 0 1; 1 -2]
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The one-to-one function h is defined below.
h(x)= 7/x-3
Find h^-1(x), where h^-1 is the inverse of h. Also state the domain and range of h in interval notation.
The inverse function h⁻¹(x) is given by: h⁻¹(x) = (7 + 3x)/x
the domain is (-∞, 3) ∪ (3, ∞).
the range is (-∞, 0) ∪ (0, ∞).
How to find the domain and rangeTo find the inverse of the function h(x) = 7/(x - 3),
y = 7/(x - 3)
swap the variables x and y:
x = 7/(y - 3)
Solve the equation for y
Multiply both sides of the equation by (y - 3):
x(y - 3) = 7
xy - 3x = 7
xy = 7 + 3x
y = (7 + 3x)/x
So, the inverse function h⁻¹(x) is given by:
h⁻¹(x) = (7 + 3x)/x
the domain and range of the original function h(x) = 7/(x - 3):
Domain: Since the denominator cannot be equal to zero, the domain of h(x) is all real numbers except x = 3. In interval notation, the domain is (-∞, 3) ∪ (3, ∞).
Range: To find the range, we need to consider the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, h(x) approaches 0, and as x approaches negative infinity, h(x) approaches 0 as well. Therefore, the range of h(x) is all real numbers except 0. In interval notation, the range is (-∞, 0) ∪ (0, ∞).
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Find the coordinate vector of p relative to the basis S = P₁ P2 P3 for P2. p = 2 - 7x + 5x²; p₁ = 1, P₂ = x, P₂ = x². (P) s= (i IM IN ).
The coordinate vector of p relative to the basis S = P₁ P₂ P₃ for P₂ is [2, -7, 5].
We are given the following:$$p = 2 - 7x + 5x^2$$$$P₁ = 1$$$$P₂ = x$$$$P₃ = x²$$
We are to find the coordinate vector of p relative to the basis S = P₁ P₂ P₃ for P₂.
First, we have to express p in terms of the basis vectors.
We can write it as:$$p = p₁P₁ + p₂P₂ + p₃P₃$$$$p = a₁(1) + a₂(x) + a₃(x²)$$
We have to find the values of a₁, a₂, and a₃.
For that, we need to equate the coefficients of p with the basis vectors.
Thus, we get:$$p = a₁(1) + a₂(x) + a₃(x²)$$$$2 - 7x + 5x² = a₁(1) + a₂(x) + a₃(x²)$$
Equating the coefficients of 1, x, and x², we get:$$a₁ = 2$$$$a₂ = -7$$$$a₃ = 5$$
Thus, the coordinate vector of p relative to the basis S = P₁ P₂ P₃ for P₂ is [2, -7, 5]
The coordinate vector of p relative to the basis S = P₁ P₂ P₃ for P₂ is [2, -7, 5].
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Random lift stops. Four students enter the lift of the five-storey building. Assume that each of them exits uniformly at random at any of five levels and independently of each other. In this question we study the random variable Z, which is the total number of lift stops (you may want to re-use some calculations from Question 3 but then you need to explain the connection). (a) Describe the sample space for this random process. (b) Find the probability that the lift stops at a fixed level i E {1, 2, 3, 4, 5). Let X, be the random variable that equals 1 if the lift stops at level i and 0, otherwise. Compute EX;. (c) Express Z in terms of X1,..., X5. Find EZ using the linearity of the expectation. (d) Find the probability that the lift stops at both levels i and j for i, j = {1, 2, 3, 4, 5). Compute EX;X;. (e) Are the variables X1 and X, independent? Justify your answer. (f) Compute EZ2 using the formula (X1 + ... + X3)2 = x;X; (where the sum is over (ij) all ordered pairs (i, j) of numbers from {1,2,3,4,5} and the linearity of the expectation. Find the variance Var Z. (g) Find the distribution of Z. That is, determine the probabilities of events Z = i for each i = 1,...,4. Compute EZ and EZ2 directly by the definition of expectation. Your answer should be in agreement with (6) and (d)
(a) The sample space for this random process can be described as the set of all possible outcomes for each of the four students exiting the lift independently at one of the five levels. Each outcome can be represented by a sequence of four numbers, where each number corresponds to the level at which a particular student exits the lift. For example, a possible outcome could be (2, 1, 4, 3), indicating that the first student exits at level 2, the second student exits at level 1, the third student exits at level 4, and the fourth student exits at level 3.
(b) To find the probability that the lift stops at a fixed level i, we need to consider each student's exit level independently. Since each student exits uniformly at random at any of the five levels, the probability that a particular student exits at level i is 1/5. Therefore, the random variable Xi follows a Bernoulli distribution with p = 1/5. The expected value of Xi, denoted as E(Xi), is equal to the probability of success, which in this case is 1/5.
(c) The total number of lift stops, Z, can be expressed as the sum of the indicator variables X1, X2, X3, X4, and X5, where Xi equals 1 if the lift stops at level i and 0 otherwise. Therefore, Z = X1 + X2 + X3 + X4 + X5. By the linearity of expectation, we have EZ = E(X1) + E(X2) + E(X3) + E(X4) + E(X5). Since each Xi follows a Bernoulli distribution with p = 1/5, the expected value of each Xi is 1/5. Thus, EZ = 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1.
(d) To find the probability that the lift stops at both levels i and j, where i and j are distinct levels from {1, 2, 3, 4, 5}, we need to consider the probabilities of each student exiting at level i and level j. Since the events are independent, the probability of the lift stopping at both levels i and j is equal to the product of the probabilities for each student. Therefore, P(Xi = 1 and Xj = 1) = (1/5) * (1/5) = 1/25. The expected value of the product of Xi and Xj, denoted as E(XiXj), is equal to the probability P(Xi = 1 and Xj = 1), which in this case is 1/25.
(e) The variables X1 and X2 are independent if the probability of their joint occurrence is equal to the product of their individual probabilities. In this case, P(X1 = 1 and X2 = 1) = P(X1 = 1) * P(X2 = 1) = (1/5) * (1/5) = 1/25. Therefore, X1 and X2 are independent. The same reasoning can be applied to show that any pair of distinct Xi and Xj are independent.
(f) To compute EZ^2, we can use the formula (X1 + X2 + X3 + X4 + X5)^2 = X1^2 + X2^2 + X3^2 + X4^2 + X5^2 + 2(X1X2 + X1X3 + X1X4 + X1X5 + X2X3 + X2X4 + X2X5 + X3X4 + X3X5 + X4X5). Using the linearity of expectation, we have EZ^2 = E(X1^2) + E(X2^2) + E(X3^2) + E(X4^2) + E(X5^2) + 2(E(X1X2) + E(X1X3) + E(X1X4) + E(X1X5) + E(X2X3) + E(X2X4) + E(X2X5) + E(X3X4) + E(X3X5) + E(X4X5)). Since each Xi follows a Bernoulli distribution, we have E(Xi^2) = Var(Xi) + (E(Xi))^2 = (1/5)(4/5) + (1/5)^2 = 9/25. Also, E(XiXj) = P(Xi = 1 and Xj = 1) = 1/25 for distinct i and j. Substituting these values, we get EZ^2 = (5 * 9/25) + (2 * 10 * 1/25) = 9/5.
To find the variance of Z, we can use the formula Var(Z) = EZ^2 - (EZ)^2. Since EZ = 1, we have Var(Z) = 9/5 - (1^2) = 4/5.
(g) The distribution of Z can be found by determining the probabilities of each event Z = i for i = 1, 2, 3, 4. Since the sample space consists of all possible outcomes of four students exiting the lift independently at any of the five levels, the values that Z can take are 0, 1, 2, 3, 4, and 5. The probabilities can be computed directly based on these outcomes, taking into account the randomness of the students' exits and the fact that each outcome is equally likely. Specifically, P(Z = i) is the probability of the lift making exactly i stops. For example, P(Z = 0) is the probability that the lift doesn't make any stops, which occurs when all four students exit at the same level. Similarly, P(Z = 1) is the probability that the lift makes exactly one stop, which occurs when three students exit at one level and one student exits at another level, or when two students exit at one level and two students exit at another level, and so on. By calculating these probabilities for each i, you can determine the distribution of Z. The expected value of Z, EZ, can be computed as the weighted sum of the possible values of Z using their respective probabilities.
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