Probability Distributions for Discrete Random Variables

A life insurance policy pays 1000 upon the death of a policyholder provided that the policyholder su at least one year but less than five years after purchasing the policy. Let X denote the number of yea policyholder survives after purchasing the policy with the following probabilities:

X 1 2 3 4 5
P(x). 0.05 0.12 0.21 0.33 0.48

Calculate the mean, the variance and the standard deviation of the payment made under this polic

To find a function y(x) that satisfies the differential equation 22y'(x)/(In(x+1)V1 - 12(1+3))+1/(1+1) and initial condition y(0)=5, we first need to separate the variables and integrate both sides.

Starting with the differential equation:

22y'(x)/(In(x+1)V1 - 12(1+3))+1/(1+1) = 0

We can rearrange to get:

22y'(x) = -1/(1+1) * (In(x+1)V1 - 12(1+3))

Dividing both sides by 22 and integrating with respect to x, we get:

y(x) = (-1/22) * (In(x+1)V1 - 12(1+3)) + C

To solve for the constant C, we can use the initial condition y(0) = 5:

y(0) = (-1/22) * (In(0+1)V1 - 12(1+3)) + C

Simplifying:

5 = (-1/22) * (In(V1) - 12(4)) + C

5 = (-1/22) * (In(V1) - 48) + C

C = 5 + (1/22) * (In(V1) - 48)

Plugging in the value of C, we get the final solution:

y(x) = (-1/22) * (In(x+1)V1 - 12(1+3)) + 5 + (1/22) * (In(V1) - 48)

This function satisfies the given differential equation and initial condition.

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A symmetric distribution the proportion of students who scored within 3 standard deviations of the mean is approximately 68% or more.

The proportion of students who scored within 3 standard deviations of the mean, we need to use the empirical rule, also known as the 68-95-99.7 rule. However, since the distribution is stated to be not bell-shaped, we cannot strictly rely on this rule. Nonetheless, we can make an approximation assuming the distribution is roughly symmetric.

According to the empirical rule, for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

The average score is 76, and the standard deviation is 5. So, within three standard deviations of the mean, we have:

Lower limit = mean - 3 * standard deviation

Upper limit = mean + 3 * standard deviation

Lower limit = 76 - 3 * 5 = 76 - 15 = 61

Upper limit = 76 + 3 * 5 = 76 + 15 = 91

Therefore, we can approximate that the proportion of students who scored within 3 standard deviations of the mean is roughly the proportion of students who scored between 61 and 91.

Since the distribution is not specified further, we cannot determine the exact proportion. However, we can approximate it by assuming a symmetric distribution. Therefore, the proportion of students who scored within 3 standard deviations of the mean is approximately 68% or more.

To find the proportion of students who scored within 3 standard deviations of the mean, we need to use the empirical rule, also known as the 68-95-99.7 rule. However, since the distribution is stated to be not bell-shaped, we cannot strictly rely on this rule. Nonetheless, we can make an approximation assuming the distribution is roughly symmetric.

According to the empirical rule, for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

In this case, the average score is 76, and the standard deviation is 5. So, within three standard deviations of the mean, we have:

Lower limit = mean - 3 ×standard deviation

Upper limit = mean + 3 × standard deviation

Lower limit = 76 - 3 × 5 = 76 - 15 = 61

Upper limit = 76 + 3 × 5 = 76 + 15 = 91

Therefore, we can approximate that the proportion of students who scored within 3 standard deviations of the mean is roughly the proportion of students who scored between 61 and 91.

Since the distribution is not specified further, we cannot determine the exact proportion. However, we can approximate it by assuming a symmetric distribution. Therefore, the proportion of students who scored within 3 standard deviations of the mean is approximately 68% .

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Answer: 6.4031

Step-by-step explanation:

d = √((x2 - x1)2 + (y2 - y1)2)

Find the difference between coordinates:

(x2 - x1) = (-2 - -6) = 4

(y2 - y1) = (-3 - 2) = -5

Square the results and sum them up:

(4)2 + (-5)2 = 16 + 25 = 41

Now Find the square root and that's your result:

Exact solution: √41 = √41

Approximate solution: 6.4031

Hope it helped

Answer:

I’d say the two people that can’t sit next to each other but to choose a different seat, if they find themselves seated next to each other they could ask someone next to them to switch seats with them. The 7 people can sit anywhere but the 2 particular people could sit away from the other.

Step-by-step explanation:

I hope I helped! ^.^’

the solution to the initial-value problem is:

x(t) = (-4.5e^(-t) + 2.5e^(t), 5e^(t)).

To solve the initial-value problem x' = (-1 1)x, x(0) = (-2, 5), we can use the matrix exponential method.

First, we find the eigenvalues and eigenvectors of the coefficient matrix (-1 1):

| -1  1 |
|       | = (λ + 1)(λ - 1) = 0
|  0  -1|

The eigenvalues are λ = -1 and λ = 1.

For λ = -1, we have:

| 0 1 |
|     | v = 0
| 0 0 |

This gives us the eigenvector v1 = (1, 0).

For λ = 1, we have:

| -2 1 |
|      | v = 0
|  0 0 |

This gives us the eigenvector v2 = (1, 2).

We can then write the general solution as:

x(t) = c1 * e^(-t) * v1 + c2 * e^(t) * v2

where c1 and c2 are constants to be determined from the initial condition x(0) = (-2, 5).

Substituting t = 0 and equating coefficients, we get:

x(0) = c1 * v1 + c2 * v2
(-2, 5) = c1 * (1, 0) + c2 * (1, 2)
-2 = c1 + c2
5 = 2c2

Solving for c1 and c2, we get:

c1 = -2 - c2 = -2 - (5/2) = -9/2
c2 = 5/2

Therefore, the solution to the initial-value problem is:

x(t) = (-9/2) * e^(-t) * (1, 0) + (5/2) * e^(t) * (1, 2)

Simplifying this expression, we get:

x(t) = (-9/2) * (e^(-t), 0) + (5/2) * (e^(t), 2e^(t))
    = (-4.5e^(-t) + 2.5e^(t), 5e^(t))

So the solution is x(t) = (-4.5e^(-t) + 2.5e^(t), 5e^(t)).

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There are 90 ways to choose the students who compete first and second in the spelling bee.

Since the order of choosing students matters in this question, we need to calculate the number of permutations. 10 students are participating in the spelling bee, and we need to choose 2 of them for the first and second place. The first student can be chosen in 10 ways, and the second student can be chosen in 9 ways (since we cannot choose the same student twice). Therefore, the number of ways to choose first and second place in the spelling bee is:

Number of ways = 10 x 9 = 90

Therefore, there are 90 ways to choose the students who compete first and second in the spelling bee.

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Jack has a 7.5 meter (750 cm) rope, gives his sister a 150 cm piece, and cuts the remaining 600 cm into 10 equal sections, with each section being 60 cm long.

Jack's rope is 7.5 meters long, which is equal to 750 centimetres. He gives his sister a piece of 150 centimetres, which leaves him with 600 centimetres of rope.
Jack then cuts the remaining piece into 10 equal sections. To find the length of each section, we can divide the total length of the rope (600 cm) by the number of sections (10):
600 cm ÷ 10 sections = 60 cm per section
Therefore, each section of rope that Jack cuts will be 60 centimetres long.

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Answer:

20

Step-by-step explanation:

h (x) = - 2x + 12

h (-4) = - 2(-4) + 12

        = 8 + 12

h (-4) = 20

The dollar price of the basket of goods in Mexico is $60. To find the dollar price of the basket of goods in Mexico,

we need to convert the price from pesos to dollars using the given exchange rate. We can do this by dividing the price in pesos by the exchange rate:

300 pesos ÷ 5 pesos/$1 = $60

Therefore, the dollar price of the basket of goods in Mexico is $60. It's important to note that exchange rates can fluctuate over time, which can impact the relative prices of goods between countries.

In this example, a weaker peso relative to the dollar makes the basket of goods appear cheaper in Mexico than in the United States.

However, if the exchange rate were to change, the relative prices of goods would also change.

Additionally, other factors such as tariffs, taxes, and transportation costs can also impact the prices of goods in different countries.

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A terminology that is described as a line, segment, or ray that bisects a segment at a right angle include the following: B. Perpendicular bisector.

In Mathematics and Geometry, a perpendicular bisector simply refers to a line, segment, or ray that bisects or divides a line segment exactly into two (2) equal halves and forms an angle that has a magnitude of 90 degrees at the point of intersection.

This ultimately implies that, a perpendicular bisector can be used to bisects or divides a line segment exactly into two (2) equal halves, in order to forms a right angle with an angle that has a magnitude of 90 degrees at the point of intersection.

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Complete Question:

Which of the following is described as a line, segment, or ray that bisects a segment at a right angle?

A. Slope

B. Perpendicular bisector

C. Midpoint

D. Angle bisector

The probability that the fair coin was selected given that the sequence HHTH was observed is 81/145.

Let F denote the event that the fair coin is chosen and let B denote the event that the biased coin is chosen. We want to find the probability of F given that the four flips of the chosen coin resulted in the sequence HHTH:

P(F|HHTH) = [[tex]\frac{P(HHTH|F) P(F)}{P(HHTH|F) P(F) + P(HHTH|B) P(B)}[/tex]]

We know that P(F) = 1/2 and P(B) = 1/2 since Beatrice selected one of the coins at random.

Next, we need to calculate the probabilities of the outcomes HHTH for each of the two coins:

P(HHTH|F) = (1/2)⁴ = 1/16

P(HHTH|B) = (2/3)² (1/3)² = 4/81

Substituting these values, we get:

P(F|HHTH) = (1/16) (1/2) / [(1/16)(1/2) + (4/81)(1/2)]

= (1/16) (1/2) / (1/2) [(1/16) + (4/81)]

= (1/16) / [1/16 + 4/81]

= 81/145

Therefore, the probability that the fair coin was selected given that the sequence HHTH was observed is 81/145.

The correct answer is not one of the given options.

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The closed-form solution for the given recurrence relation t(n) = t(n-1) * 3 with the base case t(0) = 1 is:

t(n) = 3^n


1. Start with the given recurrence relation: t(n) = t(n-1) * 3
2. Notice that the base case is t(0) = 1
3. We can rewrite the relation for a few terms to recognize the pattern:

  t(1) = t(0) * 3 = 3^1
  t(2) = t(1) * 3 = (3^1) * 3 = 3^2
  t(3) = t(2) * 3 = (3^2) * 3 = 3^3

4. Based on this pattern, we can generalize the closed-form solution as t(n) = 3^n


The closed-form solution for the given recurrence relation t(n) = t(n-1) * 3 with the base case t(0) = 1 is t(n) = 3^n.

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The derivative of F(x) is 2x times the value of f at x^2.

The problem asks to find the derivative of the function F(x) defined by an integral with respect to the variable x. The fundamental theorem of calculus relates the integral of a function over an interval to the antiderivative of the function evaluated at the endpoints of the interval.

In this case, we have:

F(x) = ∫ x^2 0 f(t) dt

By the fundamental theorem of calculus, we can take the derivative of F(x) by differentiating the integrand with respect to x:

F'(x) = d/dx [∫ x^2 0 f(t) dt]

Using the chain rule of differentiation, we can write:

F'(x) = f(x^2) * d/dx [x^2] - f(0) * d/dx [0]

The second term is zero because it's a constant. The first term can be simplified using the power rule of differentiation:

F'(x) = 2x * f(x^2)

Therefore, the derivative of F(x) is given by F'(x) = 2x * f(x^2).

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Let's denote the distance from Springfield to Teton as "D" and Bill's original rate of speed as "R" (in miles per hour). We know that at his original speed, he can travel from Springfield to Teton in 6 hours.

So, we can express this relationship as: D = R x6. Now, when Bill increases his speed by 20 mph, he can make the trip in 4 hours. So, we can express this new relationship as: D = (R + 20) x 4. Since both equations represent the distance from Springfield to Teton, we can set them equal to each other: Rx6 = (R + 20) x4 . Now, let's solve for R:

6R = 4R + 80  

2R = 80

R = 40 mph

Now that we know Bill's original rate of speed, we can calculate the distance from Springfield to Teton using either equation. Let's use the first one:
D = R x6
D = 40 x 6
D = 240 miles
So, the distance from Springfield to Teton is 240 miles.

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Answer:

6. Inverse

7. Direct

8.

a. 21

b. '10, '12

c. Inverse

9.

a. 3000

b. 6000

c. Month 9 (January)

Step-by-step explanation:

3/4 + (1/3 divided by 1/6) - (-1/2) = 3.

To solve this expression, we need to follow the order of operations: first, we simplify the expression inside the parentheses, then we perform any multiplication or division operations from left to right, and finally, we perform any addition or subtraction operations from left to right.

Let's start:

Simplify the expression inside the parentheses:

1/3 divided by 1/6 = (1/3) x (6/1) = 2

Rewrite the original expression with the simplified expression:

3/4 + 2 - (-1/2)

Solve the expression inside the parentheses:

-(-1/2) = 1/2 (double negative becomes a positive)

Rewrite the expression again with the simplified expression:

3/4 + 2 + 1/2

Convert all the fractions to a common denominator, which is 4:

3/4 + (2 x 4/4) + (1/2 x 2/2 x 2/2 x 2/2)

= 3/4 + 8/4 + 4/16

Add the fractions together:

3/4 + 8/4 + 1/4

= 12/4

= 3

Therefore, 3/4 + (1/3 divided by 1/6) - (-1/2) = 3.

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Answer:

56°

Step-by-step explanation:

angles on a straight line add up to 180°

so we are already given that the other angle which lies on the same straight line with angle X is 124 ° .which means that we should subtract 124° from 180°

The work done by the force field on the particle moving along the given helix is 60π units of work.

To find the work done by the force field F on the particle that moves along the helix r, we use the formula:

W = ∫ F · dr

where · denotes the dot product, and dr is the differential displacement vector along the path of the particle.

First, we need to calculate dr. Since the particle moves along the helix r, we can write:

dr = dx i + dy j + dz k

where dx, dy, and dz are the differentials of x, y, and z with respect to t, respectively. We have:

dx = -2sin(t) dt

dy = 2cos(t) dt

dz = 5 dt

Therefore, we can write:

dr = (-2sin(t) i + 2cos(t) j + 5k) dt

Next, we need to calculate F · dr. We have:

F · dr = (6x i + 6y j + 2k) · (-2sin(t) i + 2cos(t) j + 5k) dt

= -12sin(t) + 12cos(t) + 10 dt

Finally, we can integrate F · dr over the interval 0 ≤ t ≤ 2π to obtain the work done by the force field F on the particle that moves along the helix r:

W = ∫ F · dr = ∫ (-12sin(t) + 12cos(t) + 10) dt

= [-12cos(t) + 12sin(t) + 10t]0[tex]^(2π)[/tex]

= (-12cos(2π) + 12sin(2π) + 10(2π)) - (-12cos(0) + 12sin(0) + 10(0))

= 20π

Therefore, the work done by the force field F on the particle that moves along the helix r is 20π.

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Answer:

145°

Step-by-step explanation:

because opposite angles r equal

The probability of selecting ko and vz but excluding pfe and ge from a set of 8 stocks is 0.0625 or 6.25%.

To see why, we can use the formula for the probability of an event occurring, which is:

P(event) = (number of ways the event can occur) / (total number of possible outcomes)

The total number of possible outcomes when selecting 4 stocks from a set of 8 is:

C(8,4) = 8! / (4! * 4!) = 70

where C(n,r) is the number of combinations of r objects chosen from a set of n objects.

To count the number of ways to select ko and vz while excluding pfe and ge, we need to choose 2 stocks out of the remaining 4, which can be done in:

C(4,2) = 4! / (2! * 2!) = 6

ways.

Therefore, the probability of selecting ko and vz but excluding pfe and ge is:

P(ko and vz but not pfe or ge) = 6 / 70 = 0.0625 or 6.25%.

In summary, the probability of selecting ko and vz but excluding pfe and ge from a set of 8 stocks is 0.0625 or 6.25%, which can be calculated using the formula for probability and the number of possible outcomes that satisfy the given conditions.

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complete question:

If you selected four of these stocks at random, what is the probability that your selection included KO and VZ but excluded PFE and GE?

The game's outcomes can vary quite a bit from the expected value, and you could win more or less than $1.50 in any game.

The expected value is calculated as the sum of the product of each possible outcome and its . In this game, the possible outcomes are $3 and $0, and the probability of each outcome is 1/2 (assuming a fair coin). Therefore, the expected value of the game is:

Expected value = ($3 x 1/2) + ($0 x 1/2) = $1.50

This means that if you played the game many times, you could expect to win an average of $1.50 per game.

The variance of the game is a measure of how much the outcomes vary from the expected value. It is calculated as the sum of the squared difference between each outcome and the expected value, weighted by their respective probabilities. In this game, the variance is:

Variance = [(($3 - $1.50)^2 x 1/2) + (($0 - $1.50)^2 x 1/2)] = $2.25

This means that the outcomes of the game can vary quite a bit from the expected value, and you could win more or less than $1.50 in any given game.

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To determine whether the assumption of constant error variance is satisfied for a model with tut, independent variables x, and x₂, you would examine the residual plot where the residuals are plotted against predicted values y.

This plot is also known as the plot of residuals versus fitted values. In this plot, if the residuals are randomly scattered around the horizontal line of zero, then the assumption of constant error variance is satisfied. However, if there is a pattern in the residuals, such as a funnel shape or a curve, then the assumption of constant error variance may not be met. It is important to ensure that the assumption of constant error variance is met, as violation of this assumption can lead to biased and inefficient estimates of the model parameters. Additionally, it can affect the reliability of statistical inferences and lead to incorrect conclusions.
In summary, to determine whether the assumption of constant error variance is satisfied for a model with tut, independent variables x, and x₂, you would examine the residual plot where the residuals are plotted against predicted values y. It is important to check this assumption to ensure the validity of the model and the accuracy of the results.This plot allows you to assess the variance of the residuals and identify any patterns, which could indicate that the assumption of constant error variance may not be met. If the plot shows no discernible pattern and the spread of residuals appears to be uniform across the range of predicted values, the assumption of constant error variance is likely satisfied.

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Shawntell ran a total of 4,400,000 feet or 1,466,666.67 yards in 666 days of training for the relay race. To convert 444,000 feet to yards, we need to divide by 3 again since 1 yard is equal to 3 feet. So 1,332,000 feet ÷ 3 feet/yard + 148,000 yards = 1,466,666.67 yards

To convert 2,000 feet to yards, we need to divide by 3 since 1 yard is equal to 3 feet. So, 2,000 feet is equal to 666.67 yards.

To find out how many yards Shawntell ran in total, we can multiply 2,000 feet by 666 days, which gives us:

2,000 feet/day x 666 days = 1,332,000 feet

To convert 1,332,000 feet to yards, we need to divide by 3 again since 1 yard is equal to 3 feet. So, 1,332,000 feet is equal to 444,000 yards.

However, we need to remember that Shawntell ran 2,000 feet per day, not per yard. So, we need to divide 444,000 yards by 2,000 to find out how many days Shawntell trained for:

444,000 yards ÷ 2,000 feet/day = 222 days

This means that Shawntell ran a total of 2,000 feet x 222 days = 444,000 feet.

To convert 444,000 feet to yards, we need to divide by 3 again since 1 yard is equal to 3 feet. So, Shawntell ran a total of:

444,000 feet ÷ 3 feet/yard = 148,000 yards

Adding this to the previous calculation, we get:

1,332,000 feet ÷ 3 feet/yard + 148,000 yards = 1,466,666.67 yards

Therefore, Shawntell ran a total of 1,466,666.67 yards in 666 days of training for the relay race.

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The value of the expression is sin(E[w]) = -1 is larger than E[sin(w)] = 2/π.

We need to find whether E[sin(w)] or sin(E[w]) is larger.

Using Jensen's inequality, which states that for a convex function g, E[g(x)] >= g(E[x]), we can say:

E[sin(w)] = ∫ sin(w) * f(w) dw

Where f(w) is the probability density function of w

Taking g(x) = sin(x), which is a concave function, and using Jensen's inequality, we can say:

sin(E[w]) >= E[sin(w)]

Therefore, sin(E[w]) is larger than E[sin(w)].

Now, let's compute these two numbers:

E[sin(w)] = ∫ sin(w) * f(w) dw = ∫ sin(w) * 1/(2π - π) dw = 1/π * [(-cos(w))]π 2π = (cos(π) - cos(2π))/π = 2/π

sin(E[w]) = sin(E[w]) = sin(3π/2) = -1

Therefore, sin(E[w]) = -1 is larger than E[sin(w)] = 2/π.

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The given figure is a Triangular prism

As we know, the volume of the prism is:

[tex]V = \frac{1}{2} *l*b*h\\[/tex]

where,

l = perpendicular length of the base triangle

b = base length of the base triangle

h = height of the prism

we have given:

l = 7m, b = 24m and h = 22m

So, the Volume of the given figure is:

[tex]Volume = \frac{1}{2}*7*24*22 = 12*7*22 = 1848m^3[/tex]

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If factory produces sheets of metal that are 0.032 inch thick, the box is not deep enough to hold 400 sheets of metal.

To determine whether a box that is 12 inches deep can hold 400 sheets of metal that are 0.032 inch thick, we need to calculate the total thickness of the sheets and compare it to the depth of the box.

The total thickness of 400 sheets can be calculated by multiplying the thickness of one sheet by the number of sheets:

0.032 inch/sheet x 400 sheets = 12.8 inches

So the total thickness of 400 sheets is 12.8 inches, which is greater than the depth of the box, which is 12 inches. This means that the box is not deep enough to hold 400 sheets of metal.

If the factory wants to store 400 sheets in a box that is 12 inches deep, they would need to use sheets that are thinner than 0.032 inch, or use a box that is deeper than 12 inches.

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For distances of 45 miles or less, Company A is cheaper, while for distances greater than 45 miles, Company B is the cheaper option.

To determine at what mileage Company A charges less than Company B, we can set up an equation and solve for the variable, which in this case will represent the number of miles driven. Let x be the number of miles driven, and let C(x) represent the cost of renting a car from Company B after driving x miles.

We know that Company A charges a flat fee of $82 for unlimited mileage, so we can represent the cost of renting from Company A as a constant function C(x) = 82. For Company B, the cost function is given by:

C(x) = 55 + 0.60x

We want to find the value of x for which Company A charges less than Company B. In other words, we want to find the point at which the two cost functions intersect. To do this, we can set the two functions equal to each other and solve for x:

82 = 55 + 0.60x

27 = 0.60x

x = 45

Therefore, when the number of miles driven is 45 or less, Company A charges less than Company B. For any mileage greater than 45, it is cheaper to rent from Company B.

In summary, Company A charges a flat rate of $82 for unlimited mileage, while Company B charges an initial fee of $55 and an additional $0.60 for every mile driven. To find the point at which Company A charges less than Company B, we set the two cost functions equal to each other and solve for the number of miles driven. The result is 45 miles, meaning that for any distance of 45 miles or less, it is cheaper to rent from Company A, while for any distance greater than 45 miles, Company B is the cheaper option.

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Part A:

To find the value of m(n(2)), we need to first find the value of n(2) and then use that value to find m.

n(2) = 1/4(2)^2 - 2(2) + 4

    = 1/4(4) - 4 + 4

    = 1 - 4 + 4

    = 1

So, n(2) = 1.

Now, we can find m(1) using the equation for m:

m(1) = 3 - 2(1) + 4

    = 5

Therefore, m(n(2)) = m(1) = 5.

To find the value of n(m(1)), we need to first find the value of m(1) and then use that value to find n.

m(1) = 3 - 2(1) + 4

    = 5

So, m(1) = 5.

Now, we can find n(5) using the equation for n:

n(5) = 1/4(5)^2 - 2(5) + 4

    = 1/4(25) - 10 + 4

    = 6.25 - 10 + 4

    = 0.25

Therefore, n(m(1)) = n(5) = 0.25.

Part B:

To find the value of n(m(4)), we need to first find the value of m(4) and then use that value to find n.

m(4) = 3 - 2(4) + 4

    = -1

So, m(4) = -1.

Now, we can find n(-1) using the equation for n:

n(-1) = 1/4(-1)^2 - 2(-1) + 4

     = 1/4(1) + 2 + 4

     = 1.25 + 2 + 4

     = 7.25

Therefore, n(m(4)) = n(-1) = 7.25.

The answer is not one of the options provided.

Part C:

The functions m and n are inverse functions if and only if applying them in either order gives the identity function, i.e., m(n(x)) = x and n(m(x)) = x for all x in the domain of the functions.

From our calculations in Part A, we know that m(n(2)) = 5 and n(m(1)) = 0.25, which means that m(n(x)) ≠ x and n(m(x)) ≠ x for some values of x in the domain of the functions. Therefore, we can conclude that functions m and n are not inverse functions.

Therefore,  The pdf of e^(−x) for x ∼ expo(1) is f(x) = e^(−x) for x ≥ 0. The pdf is a decreasing function that approaches zero as x increases.

The probability density function (pdf) of an exponential distribution with parameter λ is f(x) = λe^(−λx) for x ≥ 0. In this case, λ = 1, so the pdf of e^(−x) for x ∼ expo(1) is f(x) = e^(−x) for x ≥ 0. This means that the probability of observing a value of e^(−x) between a and b is given by the integral of e^(−x) from a to b, which is equal to e^(−a) − e^(−b). The graph of this pdf shows that it is a decreasing function that approaches zero as x increases.

Therefore,  The pdf of e^(−x) for x ∼ expo(1) is f(x) = e^(−x) for x ≥ 0. The pdf is a decreasing function that approaches zero as x increases.

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The numerical value of x in the angles is 12.

The sum of angles of a straight line always add to 180 degrees.

From the diagram:

Angle 1 = ( 10x - 20 ) degrees

Angle 2 = ( 6x + 8 ) degrees

x = ?

Since angl 1 and angle 1 are on a straight line, their sum will give 180 degrees.

Hence:

Angle 1 + angle 2 = 180

Plug in the values:

( 10x - 20 ) + ( 6x + 8 ) = 180

Solve for x.

Collect and add like terms

10x + 6x -20 + 8 = 180

16x - 12 = 180

16x = 180 + 12

16x = 192

Divide both sides by 16

x = 192/16

x = 12

Therefore, x has a value of 12.

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