Let the joint pmf f(x, y) of X and Y be given by the following: (x, y) f(x, y) (1, 1)

Let \(f_{X}(x)=1 / 10, x=0,1,2, \ldots, 9\), and \(h(y \mid x)=\) \(1 /(10-x), y=x, x+1, \ldots, 9\). Find (a) \(f(x, y)\). (b) \(f_{Y}(y)\). (c) \(E(Y \mid x)\). Equation Transcription: Text Transcription: f_X(x)=1/10,x=0,1,2,,9 h(y?x)=1/(10?x),y=x,x+1,,9 f(x,y) f_Y(y) E(Y?x)

Let the joint pmf \(f(x, y)\) of \(X\) and \(Y\) be given by the following: Find the two conditional probability mass functions and the corresponding means and variances. Equation Transcription: Text Transcription: f(x, y) X Y

PROBLEM 3E Let W equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose that P(W < 1) = 0.02 and P(W > 1.072) = 0.08. Call a box of soap light, good, or heavy depending on whether {W < 1}, {1 ? W ? 1.072}, or {W > 1.072}, respectively. In n = 50 independent observations of these boxes, let X equal the number of light boxes and Y the number of good boxes. (a) What is the joint pmf of X and Y? (b) Give the name of the distribution of Y along with the values of the parameters of this distribution. (c) Given that X = 3, how is Y distributed conditionally? (d) Determine E(Y |X = 3). (e) Find ?, the correlation coefficient of X and Y.

An insurance company sells both homeowners’ insurance and automobile deductible insurance. Let \(X\) be the deductible on the homeowners’ insurance and \(Y\) the deductible on automobile insurance. Among those who take both types of insurance with this company, we find the following probabilities: (a) Compute the following probabilities: \(\begin{aligned}&P(X=500), P(Y=500), P(Y=500 \mid X=500), \\&P(Y=100 \mid X=500) . \end{aligned}\) (b) Compute the means \(\mu_{X}, \mu_{Y}\), and the variances \(\sigma_{X}^{2}\), \(\sigma_{Y}^{2}\). (c) Compute the conditional means \(E(X \mid Y=100)\), \(E(Y \mid X=500)\). (d) Compute \(\operatorname{Cov}(X, Y)\). (e) Find the correlation coefficient, \(\rho\). Equation Transcription: Text Transcription: X Y P(X = 500), P(Y = 500), P(Y = 500 | X = 500), P(Y = 100 | X = 500) E(X | Y = 100), E(Y | X = 500) Cov(X, Y) rho

Let \(X\) and \(Y\) have the joint pmf \(f(x, y)=\frac{x+y}{32}, \quad x=1,2, \quad y=1,2,3,4\) . (a) Display the joint pme and the marginal pmfs on a graph like Figure 4.3-1(a). (b) Find \(g(x \mid y)\) and draw a figure like Figure 4.3-1(b), depicting the conditional pmfs for \(y=1,2,3\), and \(4\) . (c) Find \(h(y \mid x)\) and draw a figure like Figure 4.3-1(c), depicting the conditional pmfs for \(x=1\) and 2 . (d) Find \(P(1 \leq Y \leq 3 \mid X=1), P(Y \leq 2 \mid X=2)\), and \(P(X=2 \mid \bar{Y}=3)\). (e) Find \(E(Y \mid X=1)\) and \(\operatorname{Var}(Y \mid X=1)\). Equation Transcription: Text Transcription: X Y f(x,y)=x+y/32, x=1,2, y=1,2,3,4 g(x?y) y=1,2,3 4 h(y?x) x=1 2 P(1 < or = Y < or = 3?X=1), P(Y < or = 2?X=2) P(X=2?Y?=3) E(Y?X=1) Var?(Y?X=1)

Let \(X\) and \(Y\) have a trinomial distribution with \(n=2, p_{X}=1 / 4\), and \(p_{Y}=1 / 2\). (a) Give \(E(Y \mid x)\). (b) Compare your answer in part (a) with the equation of the line of best fit in Example 4.2-2. Are they the same? Why or why not? Equation Transcription: Text Transcription: X Y n=2, p_X=1/4 p_Y=1/2 E(Y?x)

Using the joint pme from Exercise 4.2-3, find the value of \(E(Y \mid x)\) for \(x=1,2,3,4\). Do the points \([x, E(Y \mid x)]\) lie on the best-fitting line? Equation Transcription: Text Transcription: E(Y|x) x = 1, 2, 3, 4 [x,E(Y|x)]

Let \(X\) and \(Y\) have a uniform distribution on the set of points with integer coordinates in \(S=\{(x, y): 0 \leq x \leq\) \(7, x \leq y \leq x+2\}\). That is, \(f(x, y)=1 / 24,(x, y) \in S\), and both \(x\) and \(y\) are integers. Find (a) \(f_{X}(x)\). (b) \(h(y \mid x)\). (c) \(E(Y \mid x)\). (d) \(\sigma_{Y \mid x}^{2}\). (e) \(f_{Y}(y)\). Equation Transcription: . . . Text Transcription: X Y S={(x,y):0< or = x < or = 7,x< or = y< or = x+2} f(x,y)=1/24,(x,y) in S x y f_X(x) h(y?x) . E(Y?x) . sigma_Y?x^2 f_Y (y)

Let X and Y have the joint pmf f(x, y) = x + y 32 , x = 1, 2, y = 1, 2, 3, 4. (a) Display the joint pmf and the marginal pmfs on a graph like Figure 4.3-1(a). (b) Find g(x | y) and draw a figure like Figure 4.3-1(b), depicting the conditional pmfs for y = 1, 2, 3, and 4. (c) Find h(y | x) and draw a figure like Figure 4.3-1(c), depicting the conditional pmfs for x = 1 and 2. (d) Find P(1 Y 3 | X = 1), P(Y 2 | X = 2), and P(X = 2 | Y = 3). (e) Find E(Y | X = 1) and Var(Y | X = 1).

Let the joint pmf f(x, y) of X and Y be given by the following: (x, y) f(x, y) (1, 1) 3/8 (2, 1) 1/8 (1, 2) 1/8 (2, 2) 3/8 Find the two conditional probability mass functions and the corresponding means and variances.

Let W equal the weight of laundry soap in a 1-kilogram box that is distributed in Southeast Asia. Suppose that P(W < 1) = 0.02 and P(W > 1.072) = 0.08. Call a box of soap light, good, or heavy depending on whether {W < 1}, {1 W 1.072}, or {W > 1.072}, respectively. In n = 50 independent observations of these boxes, let X equal the number of light boxes and Y the number of good boxes. (a) What is the joint pmf of X and Y? (b) Give the name of the distribution of Y along with the values of the parameters of this distribution. (c) Given that X = 3, how is Y distributed conditionally? (d) Determine E(Y | X = 3). (e) Find , the correlation coefficient of X and Y.

The alleles for eye color in a certain male fruit fly are (R, W). The alleles for eye color in the mating female fruit fly are (R, W). Their offspring receive one allele for eye color from each parent. If an offspring ends up with either (W, W), (R, W), or (W, R), its eyes will look white. Let X equal the number of offspring having white eyes. Let Y equal the number of white-eyed offspring having (R, W) or (W, R) alleles. (a) If the total number of offspring is n = 400, how is X distributed? (b) Give the values of E(X) and Var(X). (c) Given that X = 300, how is Y distributed? (d) Give the value of E(Y | X = 300) and the value of Var(Y | X = 300).

Let X and Y have a trinomial distribution with n = 2, pX = 1/4, and pY = 1/2. (a) Give E(Y | x). (b) Compare your answer in part (a) with the equation of the line of best fit in Example 4.2-2. Are they the same? Why or why not?

An insurance company sells both homeowners insurance and automobile deductible insurance. Let X be the deductible on the homeowners insurance and Y the deductible on automobile insurance. Among those who take both types of insurance with this company, we find the following probabilities:x y 100 500 1000 1000 0.05 0.10 0.15 500 0.10 0.20 0.05 100 0.20 0.10 0.05 (a) Compute the following probabilities: P(X = 500), P(Y = 500), P(Y = 500 | X = 500), P(Y = 100 | X = 500). (b) Compute the means X , Y , and the variances 2 X , 2 Y . (c) Compute the conditional means E(X | Y = 100), E(Y | X = 500). (d) Compute Cov(X, Y). (e) Find the correlation coefficient,

Using the joint pmf from Exercise 4.2-3, find the value of E(Y | x) for x = 1, 2, 3, 4. Do the points [x, E(Y | x)] lie on the best-fitting line? Exercise 4.2-3 Roll a fair four-sided die twice. Let X equal the outcome on the first roll, and let Y equal the sum of the two rolls. (a) Determine \(\mu_x,\mu_y,\sigma_x^2,\sigma_y^2\), Cov(X,Y) and \(\rho\). (b) Find the equation of the least squares regression line and draw it on your graph. Does the line make sense to you intuitively?

A fair six-sided die is rolled 30 independent times. Let X be the number of ones and Y the number of twos. (a) What is the joint pmf of X and Y? (b) Find the conditional pmf of X, given Y = y. (c) Compute E(X2 4XY + 3Y2).

Let X and Y have a uniform distribution on the set of points with integer coordinates in S = {(x, y): 0 x 7, x y x + 2}. That is, f(x, y) = 1/24, (x, y) S, and both x and y are integers. Find (a) fX (x). (b) h(y | x). (c) E(Y | x). (d) 2 Y | x. (e) fY (y).

Let fX (x) = 1/10, x = 0, 1, 2, ... , 9, and h(y | x) = 1/(10 x), y = x, x + 1, ... , 9. Find (a) f(x, y). (b) fY (y). (c) E(Y | x).

Choose a random integer X from the interval [0, 4]. Then choose a random integer Y from the interval [0, x], where x is the observed value of X. Make assumptions about the marginal pmf fX (x) and the conditional pmf h(y | x) and compute P(X + Y > 4).