We do not have sufficient evidence to conclude that the variance of monthly incomes is higher for male employees than it is for female employees at the 0.01 level of significance.
a) State the hypotheses:
Null Hypothesis (H0): The variance of monthly incomes for male employees is equal to or less than the variance of monthly incomes for female employees.
Alternative Hypothesis (Ha): The variance of monthly incomes for male employees is higher than the variance of monthly incomes for female employees.
b) Calculate the test statistic:
We can use the F-test to compare the variances of the two samples. The test statistic is:
[tex]F = s1^2 / s2^2[/tex]
where s1 and s2 are the sample standard deviations, and F follows an F-distribution with (n1-1) and (n2-1) degrees of freedom.
For female employees:
n1 = 61
[tex]s1 = $194.40[/tex]
[tex]s1^2 = ($194.40)^2 = $37,825.60[/tex]
For male employees:
n2 = 121
s2 = $269.92
[tex]s2^2 = ($269.92)^2 = $72,941.29[/tex]
So, the test statistic is:
[tex]F = s1^2 / s2^2 = $37,825.60 / $72,941.29 = 0.518[/tex]
c) State the rejection criterion for the null hypothesis:
We will use a significance level of 0.01. Since this is a one-tailed test (we are testing if the variance of male employees is higher than the variance of female employees), the rejection region is in the upper tail of the F-distribution. We need to find the critical value of F with (60, 120) degrees of freedom at the 0.01 level of significance. Using a statistical table or calculator, we find that the critical value is 2.74.
d) Draw your conclusion:
The calculated F-value (0.518) is less than the critical F-value (2.74). Therefore, we fail to reject the null hypothesis. We do not have sufficient evidence to conclude that the variance of monthly incomes is higher for male employees than it is for female employees at the 0.01 level of significance.
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Andre, Lin, and Noah each designed and built a paper airplane. They launched each plane several times and recorded the distance of each flight in yards. Write the five-number summary for the data for each airplane. Then, calculate the interquartile range for each data set.
Answer:
Andre's: Min = 18, Q1 = 23.5, Median = 28.5, Q3 = 31.5, Max = 35, IQR = 8
Lin's: Min = 15, Q1 = 19, Median = 21.5, Q3 = 24, Max = 33, IQR = 5
Noah's: Min = 10, Q1 = 12.5, Median = 19, Q3 = 22.5, Max = 25, IQR = 10
Step-by-step explanation:
following is the probability distribution of a random variable that represents the number of extracurricular activities a college freshman participates in.Part 1 Find the probability that a student participates in exactly two activities The probability that a student participates in exactly two activities is
The table, we cannot determine the probability of a student participating in exactly two activities.
The probability distribution table is not provided in the question, but assuming that it is a valid probability distribution, we can use it to find the probability that a student participates in exactly two activities.
Let X be the random variable representing the number of extracurricular activities a college freshman participates in, and let p(x) denote the probability of X taking the value x.
Then, we want to find p(2), the probability that a student participates in exactly two activities. This can be obtained from the probability distribution table.
Without the table, we cannot determine the probability of a student participating in exactly two activities.
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Question 5(Multiple Choice Worth 2 points)
(Properties of Operations MC)
What is an equivalent form of 15(p+ 4) - 12(2q + 4)?
15p24q+ 12
O15p -24q+8
60p-72q
-9pq
Answer:
15p - 24q +8
Step-by-step explanation:
Which of the following is equivalent to
5x²+2=-7x
The equivalent expression is (5x + 2)(x + 1)
What are algebraic expressions?Algebraic expressions are mathematical expressions that comprises of variables, terms, coefficients, factors and constants.
Also, note that equivalent expressions are defined as expressions having the same solution but differ in the arrangement of its variables.
From the information given, we have that;
5x²+2=-7x
Put into standard form
5x² + 7x + 2 = 0
To solve the quadratic equation, we have to find the pair factors of the product of 5 and 2 that sum up to 7, we have;
Substitute the values
5x² + 5x + 2x + 2 = 0
group in pairs
(5x² + 5x) + ( 2x + 2 ) = 0
factorize the expression
5x(x + 1) + 2(x + 1) = 0
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What is the mean of the data set?
A. 42
B. 45
C. 20
D. 40
The mean of the data-set in this problem is given as follows:
41.6 inches.
How to calculate the mean of a data-set?The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.
Considering the stem-and-leaf plot, the observations are given as follows:
20, 32, 34, 36, 40, 42, 44, 48, 55, 65.
The sum of the observations is given as follows:
20 + 32 + 34 + 36 + 40 + 42 + 44 + 48 + 55 + 65 = 416 inches.
There are 10 observations, hence the mean is given as follows:
416/10 = 41.6 inches.
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Let x^8+3x^4-4=p_1(x)p_2(x)...p_k(x) where each non-constant polynomial p_i(x) is monic with integer coefficients, and cannot be factored further over the integers. Compute p_1(1)+p_2(1)+...+p_k(1).
Answer: We can factor the given polynomial as follows:
x^8 + 3x^4 - 4 = (x^4 - 1)(x^4 + 4)
= (x^2 - 1)(x^2 + 1)(x^2 - 2x + 1)(x^2 + 2x + 1)
The four factors on the right-hand side are all monic polynomials with integer coefficients that cannot be factored further over the integers. Therefore, we have k = 4, and we can compute p_1(1) + p_2(1) + p_3(1) + p_4(1) as follows:
p_1(1) + p_2(1) + p_3(1) + p_4(1) = (1^2 - 1) + (1^2 + 1) + (1^2 - 2(1) + 1) + (1^2 + 2(1) + 1)
= 0 + 2 + 0 + 6
= 8
Therefore, p_1(1) + p_2(1) + p_3(1) + p_4(1) = 8.
Step-by-step explanation:
the lengths of full-grown scorpions of a certain variety have a mean of 1.96 inches and a standard deviation of 0.08 inch. assuming the distribution of the lengths has roughly the shape of a normal disribution, find the value above which we could expect the longest 20% of these scorpions.
We can expect the longest 20% of these scorpions to be above a length of approximately 2.0272 inches.
To find the value above which we could expect the longest 20% of these scorpions, we need to use the z-score formula. First, we need to find the z-score that corresponds to the 80th percentile, which is the complement of the top 20%. Using a standard normal distribution table or calculator, we find that the z-score corresponding to the 80th percentile is 0.84.
Next, we use the formula z = (x - mu) / sigma, where z is the z-score, x is the value we are trying to find, mu is the mean, and sigma is the standard deviation. We plug in the given values and solve for x:
0.84 = (x - 1.96) / 0.08
0.0672 = x - 1.96
x = 2.0272
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What is the principal that will grow to $5100 in two years,
eight months at 4.3% compounded semi-annually? The principal is
$=
The principal that will grow to $5,100 in two years, eight months at 4.3% compounded semi-annually is approximately $4,568.20.
To find the principal that will grow to $5,100 in two years and eight months at 4.3% compounded semi-annually, you can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = final amount ($5,100)
P = principal amount (what we're trying to find)
r = annual interest rate (4.3% or 0.043)
n = number of times interest is compounded per year (semi-annually, so 2)
t = time in years (2 years and 8 months or 2.67 years)
First, plug in the values:
$5,100 = P(1 + 0.043/2)^(2*2.67)
Next, solve for P:
P = $5,100 / (1 + 0.043/2)^(2*2.67)
P = $5,100 / (1.0215)^(5.34)
P = $5,100 / 1.11726707
P ≈ $4,568.20
The principal that will grow to $5,100 in two years, eight months at 4.3% compounded semi-annually is approximately $4,568.20.
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How to solve for A and Z?
The length of the missing sides of the two quadrilaterals are listed below:
a = 5z = 4.219How to find the missing lengths in quadrilaterals
In this problem we must determine the length of missing sides in two quadrilaterals, this can be done with the help of Pythagorean theorem and properties for special right triangles:
r = √(x² + y²)
45 - 90 - 45 right triangle
r = √2 · x = √2 · y
Where:
x, y - Legsr - HypotenuseNow we proceed to determine the missing sides for each case:
a = √[(6 - 3)² + 4²]
a = √(3² + 4²)
a = √25
a = 5
Case 2
z = √[(22 - 4√2 - 15)² + 4²]
z = √[(7 - 4√2)² + 4²]
z = 4.219
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(0.70 * (1 - 0.10) = 0.70 * 0.9)
Answer:
The simplified result of the expression is 0.63.
Step-by-step explanation:
you have a sample of 20 pieces of chocolate that are all of the same shape and size (5 pieces have peanuts, 5 pieces have almonds, 5 pieces have macadamia nuts, 5 pieces have no nuts). you weigh each of the 20 pieces of chocolate and get the following weights (in grams). you want to know if the weights across all types of chocolate are statistically significantly different from one another using a significance level of 0.05.
Determine if the weights across all types of chocolate are statistically significantly different from one another using a significance level of 0.05. To do this, we'll use an ANOVA (Analysis of Variance) test. Here are the steps to perform the test:
1. Organize the data: Group the weights of each type of chocolate (peanuts, almonds, macadamia nuts, and no nuts) separately.
2. Calculate the means: Find the mean weight for each group and the overall mean weight for all 20 pieces of chocolate.
3. Calculate the Sum of Squares Between (SSB) and Sum of Squares Within (SSW): SSB represents the variation between groups, and SSW represents the variation within each group.
4. Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW): Divide SSB by the degrees of freedom between groups (k-1, where k is the number of groups), and divide SSW by the degrees of freedom within groups (N-k, where N is the total number of samples).
5. Calculate the F statistic: Divide MSB by MSW.
6. Determine the critical F value: Using an F distribution table, find the critical F value corresponding to a significance level of 0.05 and the degrees of freedom between and within groups.
7. Compare the calculated F statistic to the critical F value: If the calculated F statistic is greater than the critical F value, the difference in weights across the types of chocolate is considered statistically significant.
If you follow these steps with the provided weight data, you'll be able to determine if the differences in chocolate weights are statistically significant at a 0.05 significance level.
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√175x²y³ simplify radical expression
Answer:
[tex] \sqrt{175 {x}^{2} {y}^{3} } = \sqrt{175} \sqrt{ {x}^{2} } \sqrt{ {y}^{3} } = \sqrt{25} \sqrt{ {x}^{2} } \sqrt{ {y}^{2} } \sqrt{7} \sqrt{y} = 5xy \sqrt{7y} [/tex]
The market price of a t-shirt is $15.00. It is discounted at 10% off. What is the selling price of the t-shirt?
(Enter your answer following the model, i.e. $01.01)
Answer: $13.5
Step-by-step explanation:
1. 15 x 10 =150
2. 150/100=1.5
3. 15.00 - 1.50= 13.5
The solution of the boundary value problem (D^2 +4^2)y=0,given that y(0) = 0 and y(phi/8) = 1. a) y = cos 4x, b) y = 3 sin 4x, c) y) = 4 sin 4x. d) y=sin 4x
The correct solution to the given boundary value problem (D^2 + 4^2)y = 0, with y(0) = 0 and y(phi/8) = 1, is d) y = sin 4x.
This can be found by using the value problem characteristic equation of the differential equation, which is r^2 + 16 = 0. Solving for r, we get r = +/- 4i. Therefore, the general solution is y(x) = c1 sin 4x + c2 cos 4x.
To find the values of c1 and c2, we use the boundary conditions. First, we have y(0) = 0, which gives c2 = 0. Then, we have y(phi/8) = 1, which gives c1 = 1/4. Thus, the final solution is y(x) = (1/4) sin 4x.
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The three sides of a triangle have lengths of x units,
(x-4) units, and (x² - 2x - 5) units for some value of x greater than 4. What is the perimeter, in units, of the triangle?
Answer:
The perimeter is x² - 9 units-----------------------
The perimeter is the sum of three side lengths:
P = x + (x - 4) + (x² - 2x - 5) P = x + x - 4 + x² - 2x - 5P = x² - 9(1 point) Determine whether the following series converges or diverges. (-1)n-1 (- n=1 Input C for convergence and D for divergence: Note: You have only one chance to enter your answer
The series ∑n-1 to ∞ [tex](-1)^{n-1} )/\frac{\sqrt(n}{n+5} }[/tex] is convergence (C).
The given series is:
∑n-1 to ∞ [tex](-1)^{n-1} )/\frac{\sqrt(n}{n+5} }[/tex]
To determine if the series converges or diverges, we can use the alternating series test. The alternating series test states that if a series has alternating terms that decrease in absolute value and converge to zero, then the series converges.
In this series, the terms alternate in sign and decrease in absolute value, since the denominator (n) increases as n increases. Also, as n approaches infinity, the term [tex](-1)^{n-1}[/tex]oscillates between 1 and -1, but does not converge to a specific value. However, the absolute value of the term 1/n approaches 0 as n approaches infinity.
Therefore, by the alternating series test, the given series converges. The answer is C (convergence).
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Help please!! <3
Anything would be much appreciated!!
Answer:
a) It is not possible to find the mean because these are words, not numbers.
b) If we put these words in alphabetical order, we have:
blue, blue, green, purple, purple, purple, red, red
The median word here is purple.
c) It is possible to find the mode, which in this case is the word that appears the most times in this list. That word is purple, which appears three times.
Suppose the sample space for a continuous random variable is 0 to 200. If
the area under the density graph for the variable from 0 to 50 is 0.25, then the
area under the density graph from 50 to 200 is 0.75.
OA. True
B. False
Let a > 0 be real. Consider the complex function f(z) 1 + cos az 02 22 - Identify the order of all the poles of f(z) on the finite complex plane. Evaluate the residue of f(z) at these poles.
Hi! To answer your question, let's analyze the complex function f(z) given by f(z) = 1 + cos(az)/(z^2).
First, we need to identify the poles of the function. A pole occurs when the denominator of the function is zero. In this case, the poles are at z = 0. However, the order of the pole is determined by the number of times the denominator vanishes, which is given by the exponent of z in the denominator. Here, the exponent is 2, so the order of the pole is 2.
Now, let's find the residue of complex function f(z) at the pole z = 0. To do this, we can apply the residue formula for a second-order pole:
Res[f(z), z = 0] = lim (z -> 0) [(z^2 * (1 + cos(az)))/(z^2)]'
where ' denotes the first derivative with respect to z.
First, let's find the derivative:
d(1 + cos(az))/dz = -a * sin(az)
Now, substitute this back into the residue formula:
Res[f(z), z = 0] = lim (z -> 0) [z^2 * (-a * sin(az))]
Since sin(0) = 0, the limit evaluates to 0. Therefore, the residue of f(z) at the pole z = 0 is 0.
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Castel and Kali each improved their yards by planting rose bushes and geraniums. They brought their supplies from the same store. Castel spent $115 on 5 rose bushes and 12 geraniums. Kail spent $94 on 10 rose bushes and 8 geraniums.
(a) Write a system of equations that represents the scenario
(b) Solve the system to determine the cost of one rose bush and the cost of one geraniums.
a) The system of equations that represents the scenario is given as follows:
5x + 12y = 115.10x + 8y = 94.b) The costs are given as follows:
One bush: $2.6.One geranium: 8.5.How to define the system of equations?The variables for the system of equations are defined as follows:
Variable x: cost of a bush.Variable y: cost of a geranium;Castel spent $115 on 5 rose bushes and 12 geraniums, hence:
5x + 12y = 115.
Kail spent $94 on 10 rose bushes and 8 geraniums, hence:
10x + 8y = 94
Then the system is defined as follows:
5x + 12y = 115.10x + 8y = 94.Multiplying the first equation by 2 and subtracting by the second, we have that the value of y is obtained as follows:
24y - 8y = 230 - 94
16y = 136
y = 136/16
y = 8.5.
Then the value of x is obtained as follows:
5x + 12(8.5) = 115
x = (115 - 12 x 8.5)/5
x = 2.6.
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Calculator
Here is a picture of a cube, and the net of this cube.
What is the surface area of this cube?
Enter your answer in the box.
cm²
t.
11 cm
11 cm
The surface area of the cube is 726 cm^2.
What is the surface area of a shape?The surface area of a given shape is the summation or the total value of the area of each of its external surfaces. Thus the total surface of a shape depends on the number of its external surface, and the shape of each.
In the given question, the cube has a side length of 11 cm. Since each surface of the cube is formed from a square, then;
area of a square = length x length
= 11 x 11
= 121 sq. cm.
Total surface area of the cube = number of its surface x area of each surface
= 6 x 121
= 726
The surface area of the cube is 726 sq. cm.
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In the figure there are 5 equal rectangles and each of its sides is marked with a number as indicated in the drawing. Rectangles are placed without rotating or flipping in positions I, II, III, IV, and V in such a way that the sides that stick together in two rectangles have the same number. Which of the rectangles should go in position I?
The rectangle which should go in position I is rectangle A.
We are given that;
The rectangles A,B,C and D with numbers
Now,
To take the same the number of side
If we take A on 1 place
F will be on second place
And B will be on 4th place
Therefore, by algebra the answer will be rectangle A.
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If you subtract 16 from my number and multiply the difference by -3, the result is -60
36 is the number that satisfies the given condition.
Let's say your number is represented by the variable "x". According to the problem, when you subtract 16 from your number and multiply the difference by -3, the result is -60. We can translate this into an equation as follows:
-3(x - 16) = -60
To solve for x, we'll first simplify the left-hand side of the equation using the distributive property:
-3x + 48 = -60
Next, we'll isolate the variable x by subtracting 48 from both sides of the equation:
-3x = -108
Finally, we can solve for x by dividing both sides of the equation by -3:
x = 36
Therefore, if you subtract 16 from 36 and multiply the difference by -3, the result is -60:
-3(36 - 16) = -60
-3(20) = -60
-60 = -60
So 36 is the number that satisfies the given condition.
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4. What are the median and mode of the
plant height data?
⬇️
Numbers:
13,14,15,17,17,17,19,20,21
Answer: 5
Step-by-step explanation:
Answer:17
Step-by-step explanation:
The median of this data set is 17, since if you cross one # off of both sides, you will eventually get to the middle fo the data set, pointing to 17.
Solve the system of equations.
y = −2x+4
2x+y=4
What is the solution to the system of equations?
A. No solution
B. Parallel lines
C. Infinitely many solutions
Answer:
C
Step-by-step explanation:
y=-2x+4
2x+y=4
2x-2x+4=4
4=4
you are dealt one card from a standard 52-card deck. find the probability of being dealt an ace or a 8. group of answer choices
There are 4 aces and 4 nights in a standard 52-card deck. So, the total number of cards that can be considered as a successful outcome is 8. Therefore, the probability of being dealt an ace or an 8 is 8/52 or 2/13. To find the probability of being dealt an Ace or an 8, follow these steps:
1. Identify the total number of cards in the deck: There are 52 cards in a standard deck.
2. Determine the number of Aces and 8s in the deck: There are 4 Aces and 4 eights, totaling 8 cards (4 Aces + 4 eights).
3. Calculate the probability: Divide the number of desired outcomes (Aces and 8s) by the total number of cards in the deck.
Probability = (Number of Aces and 8s) / (Total number of cards)
Probability = 8 / 52
4. Simplify the fraction: 8/52 can be simplified to 2/13.
So, the probability of being dealt an Ace or an 8 from a standard 52-card deck is 2/13.
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what is the solution for x+1*-4x+1
The solution to the product of the given equation is:
-4x² - 3x + 1
How to multiply linear equations?A linear equation is defined as an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant.
Thus, looking at the given equation, we have:
(x + 1) * (-4x + 1)
Expanding this gives:
-4x² - 4x + x + 1
-4x² - 3x + 1
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5. A random variable X has the moment generating function 0.03 Mx(0) t< - log 0.97 1 -0.97e Name the probability distribution of X and specify its parameter(s). (b) Let Y = X1 + X2 + X3 where X1, X3,
Y follows a negative binomial distribution with parameters r = 3 and p = 0.97.
The moment generating function (MGF) of a random variable X is defined as Mx(t) = E(e^(tX)).
(a) The given MGF is 0.03 Mx(0) t< - log 0.97 1 -0.97e^(tX)
The MGF of the geometric distribution with parameter p is given by Mx(t) = E(e^(tX)) = Σ [p(1-p)^(k-1)]e^(tk), where the sum is taken over all non-negative integers k.
Comparing this with the given MGF, we can see that p = 0.97. Therefore, X follows a geometric distribution with parameter p = 0.97.
(b) Let Y = X1 + X2 + X3, where X1, X3, and X3 are independent and identically distributed geometric random variables with parameter p = 0.97.
The MGF of Y can be obtained as follows:
My(t) = E(e^(tY)) = E(e^(t(X1 + X2 + X3))) = E(e^(tX1) * e^(tX2) * e^(tX3))
= Mx(t)^3, since X1, X2, and X3 are independent and identically distributed with the same MGF
Substituting the given MGF of X, we get:
My(t) = (0.03 Mx(0) t< - log 0.97 1 -0.97e^(t))^3
Therefore, Y follows a negative binomial distribution with parameters r = 3 and p = 0.97.
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fill in the price and the total, marginal, and average revenue sendit earns when it rents 0, 1, 2, or 3 trucks during move-in week.
Renting 0 trucks the Marginal Revenue (MR) = Not applicable, and Average Revenue (AR) = Not applicable. Renting 1 truck the Marginal Revenue (MR) = $P (since it's the additional revenue gained from renting 1 truck), Average Revenue (AR) = Total Revenue / Quantity = P / 1 = $P.
Renting 2 trucks Marginal Revenue (MR) = ($2P - $P) = $P (since it's the additional revenue gained from renting the second truck), Average Revenue (AR) = Total Revenue / Quantity = 2P / 2 = $P. Renting 3 trucks Marginal Revenue (MR) = ($3P - $2P) = $P (since it's the additional revenue gained from renting the third truck), Average Revenue (AR) = Total Revenue / Quantity = 3P / 3 = $P.
To help you with your question, we need to know the rental price per truck and the costs associated with renting these trucks. Since this information is not provided, I will assume a rental price of P dollars per truck. Based on this assumption, we can calculate total, marginal, and average revenue for Sendit when renting 0, 1, 2, or 3 trucks during the move-in week.
1. Renting 0 trucks:
Total Revenue (TR) = 0 * P = $0
Marginal Revenue (MR) = Not applicable
Average Revenue (AR) = Not applicable
2. Renting 1 truck:
Total Revenue (TR) = 1 * P = $P
Marginal Revenue (MR) = $P (since it's the additional revenue gained from renting 1 truck)
Average Revenue (AR) = Total Revenue / Quantity = P / 1 = $P
3. Renting 2 trucks:
Total Revenue (TR) = 2 * P = $2P
Marginal Revenue (MR) = ($2P - $P) = $P (since it's the additional revenue gained from renting the second truck)
Average Revenue (AR) = Total Revenue / Quantity = 2P / 2 = $P
4. Renting 3 trucks:
Total Revenue (TR) = 3 * P = $3P
Marginal Revenue (MR) = ($3P - $2P) = $P (since it's the additional revenue gained from renting the third truck)
Average Revenue (AR) = Total Revenue / Quantity = 3P / 3 = $P
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12. Let the continuous random vector (X, Y) have the joint pdf f(x, y) = c(x+y) over the unit square.
i. Find the value of e so that the function is a valid joint pdf.
ii. Find P(X<.5, Y <5).
iii. Find P(YX).
iii. Find P(X + Y) < 5
iv. Compute E(XY) and E(X + Y).
(i) c = 1/2 and the joint pdf is f(x, y) = (x+y)/2 over the unit square.
(ii) 1/16
(iii) 1/9
iv) 5/3
(v) E(X+Y) = 5/6.
we have,
i.
In order for f(x, y) to be a valid joint pdf, it must satisfy two conditions:
It must be non-negative for all (x,y)
The integral over the entire support must equal 1.
To satisfy the first condition, we need c(x+y) to be non-negative.
This is true as long as c is non-negative and x+y is non-negative over the support, which is the unit square [0,1]x[0,1]. Since x and y are both non-negative over the unit square, we need c to be non-negative as well.
To satisfy the second condition, we integrate f(x, y) over the unit square and set it equal to 1:
1 = ∫∫ f(x, y) dx dy
= ∫∫ c(x+y) dx dy
= c ∫∫ (x+y) dx dy
= c [∫∫ x dx dy + ∫∫ y dx dy]
= c [∫ 0^1 ∫ 0^1 x dx dy + ∫ 0^1 ∫ 0^1 y dx dy]
= c [(1/2) + (1/2)]
= c
ii.
P(X < 0.5, Y < 0.5) can be found by integrating the joint pdf over the region where X < 0.5 and Y < 0.5:
P(X < 0.5, Y < 0.5) = ∫ 0^0.5 ∫ 0^0.5 (x+y)/2 dy dx
= ∫ 0^0.5 [(xy/2) + (y^2/4)]_0^0.5 dx
= ∫ 0^0.5 [(x/4) + (1/16)] dx
= [(x^2/8) + (x/16)]_0^0.5
= (1/32) + (1/32)
= 1/16
iii.
P(Y<X) can be found by integrating the joint pdf over the region where
Y < X:
P(Y < X) = ∫ 0^1 ∫ 0^x (x+y)/2 dy dx
= ∫ 0^1 [(xy/2) + (y^2/4)]_0^x dx
= ∫ 0^1 [(x^3/6) + (x^3/12)] dx
= (1/9)
iv.
P(X+Y) < 5 can be found by integrating the joint pdf over the region where X+Y < 5:
P(X+Y < 5) = ∫ 0^1 ∫ 0^(5-x) (x+y)/2 dy dx
= ∫ 0^1 [(xy/2) + (y^2/4)]_0^(5-x) dx
= ∫ 0^1 [(x(5-x)/2) + ((5-x)^2/8)] dx
= 5/3
v.
The expected value of XY can be found by integrating the product xy times the joint pdf over the entire support:
E(XY) = ∫∫ xy f(x, y) dx dy
E(XY) = ∫∫ xy (x+y)/2 dx dy
= ∫∫ (x^2y + xy^2)/2 dx dy
= ∫ 0^1 ∫ 0^1 (x^2y + xy^2)/2 dx dy
= ∫ 0^1 [(x^3*y/3) + (xy^3/6)]_0^1 dy
= ∫ 0^1 [(y/3) + (y/6)] dy
= 1/4
The expected value of X+Y can be found by integrating the sum (x+y) times the joint pdf over the entire support:
E(X+Y) = ∫∫ (x+y) f(x, y) dx dy
= ∫∫ (x+y) (x+y)/2 dx dy
= ∫∫ [(x^2+2xy+y^2)/2] dx dy
= ∫ 0^1 ∫ 0^1 [(x^2+2xy+y^2)/2] dx dy
= ∫ 0^1 [(x^3/3) + (xy^2/2) + (y^3/3)]_0^1 dy
= ∫ 0^1 [(1/3) + (y/2) + (y^2/3)] dy
= 5/6
Thus,
(i) c = 1/2 and the joint pdf is f(x, y) = (x+y)/2 over the unit square.
(ii) 1/16
(iii) 1/9
iv) 5/3
(v) E(X+Y) = 5/6.
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