Let the joint pmf of X and Y be f(x, y) = 1/4, (x, y) S = {(0, 0), (1, 1), (1, 1), (2

Let the random variables \(X\) and \(Y\) have the joint pmf \(f(x, y)=\frac{x+y}{32}, \quad x=1,2, \quad y=1,2,3,4\) Find the means \(\mu_{X}\) and \(\mu_{Y}\), the variances \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\), and the correlation coefficient \(\rho\). Equation Transcription: , Text Transcription: X Y f(x,y)=x+y/32, x=1,2, y=1,2,3,4 mu_X mu_Y, sigma_X^2 sigma_Y^2 rho

Let \(X\) and \(Y\) have the joint pmf defined by \(f(0,0)=\) \(f(1,2)=0.2, f(0,1)=f(1,1)=0.3\). (a) Depict the points and corresponding probabilities on a graph. (b) Give the marginal pmfs in the "margins." (c) Compute \(\mu_{X}, \mu_{Y}, \sigma_{X}^{2}, \sigma_{Y}^{2}, \operatorname{Cov}(X, Y)\), and \(\rho\). (d) Find the equation of the least squares regression line and draw it on your graph. Does the line make sense to you intuitively? Equation Transcription: Text Transcription: X Y f(0, 0) = f(1, 2) = 0.2, f(0, 1) = f(1, 1) = 0.3 mu_X, mu_Y, sigma_X^2, sigma_Y^2, Cov?(X,Y) rho

Let \(X\) and \(Y\) be random variables with respective means \(\mu_{X}\) and \(\mu_{Y}\), respective variances \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\), and correlation coefficient \(\rho\). Fit the line \(y=a+b x\) by the method of least squares to the probability distribution by minimizing the expectation \(K(a, b)=E\left[(Y-a-b X)^{2}\right]\) with respect to \(a\) and \(b\). Hint: Consider \(\partial K / \partial a=0\) and \(\partial K / \partial b=0\), and solve simultaneously. Equation Transcription: , Text Transcription: X Y mu_X mu_Y, sigma_X^2 sigma_Y^2 y=a+bx K(a,b)=E[(Y-a-bX)^2] a b Partial K/Partial a=0 Partial K/Partial b=0

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 \(\mu_{X}, \mu_{Y}, \sigma_{X}^{2}, \sigma_{Y}^{2}, \operatorname{Cov}(X, Y)\), 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? Equation Transcription: Text Transcription: X Y f(0, 0) = f(1, 2) = 0.2, f(0, 1) = f(1, 1) = 0.3 mu_X, mu_Y, sigma_X^2, sigma_Y^2, Cov?(X,Y) rho

Let X and Y have a trinomial distribution with parameters n = 3, \(p_X = 1/6\), and \(p_Y = 1/2\). Find (a) E(X). (b) E(Y). (c) Var(X). (d) Var(Y). (e) Cov(X, Y). (f) \(\rho\). Note that \(\rho = - \sqrt {p_X p_Y /[(1 - p_X )(1 - p_Y )]}\) in this case. (Indeed, the formula holds in general for the trinomial distribution; see Example 4.3-3.)

PROBLEM 6E The joint pmf of X and Y is f (x, y) = 1/6, 0 ?x + y ? 2, where x and y are nonnegative integers. (a) Sketch the support of X and Y. (b) Record the marginal pmfs fX(x) and fY(y) in the “margins.” (c) Compute Cov(X,Y). (d) Determine ?, the correlation coefficient. (e) Find the best-fitting line and draw it on your figure.

Let the joint pmf of \(X\) and \(Y\) be \(f(x, y)=1 / 4, \quad(x, y) \in S=\{(0,0),(1,1),(1,-1),(2,0)\}\) (a) Are \(X\) and \(Y\) independent? (b) Calculate \(\operatorname{Cov}(X, Y)\) and \(\rho\). This exercise also illustrates the fact that dependent random variables can have a correlation coefficient of zero. Equation Transcription: Text Transcription: X Y f(x,y)=1/4, (x,y) in S={(0,0),(1,1),(1,-1),(2,0)} Cov?(X,Y) rho

A car dealer sells \(X\) cars each day and always tries to sell an extended warranty on each of these cars. (In our opinion, most of these warranties are not good deals.) Let \(Y\) be the number of extended warranties sold; then \(Y \leq X\). The joint pmf of \(X\) and \(Y\) is given by \(\begin{aligned}f(x, y) &=c(x+1)(4-x)(y+1)(3-y), \\x &=0,1,2,3, \quad y=0,1,2, \text { with } y \leq x .\end{aligned}\) (a) Find the value of \(c\). (b) Sketch the support of \(X\) and \(Y\). (c) Record the marginal pmfs \(f_{X}(x)\) and \(f_{Y}(y)\) in the "margins." (d) Are \(X\) and \(Y\) independent? (e) Compute \(\mu_{X}\) and \(\sigma_{X}^{2}\). (f) Compute \(\mu_{Y}\) and \(\sigma_{Y}^{2}\). (g) Compute \(\operatorname{Cov}(X, Y)\). (h) Determine ?, the correlation coefficient. (i) Find the best-fitting line and draw it on your figure. Equation Transcription: . Text Transcription: X Y Y < or = X f(x,y) =c(x+1)(4-x)(y+1)(3-y), x =0,1,2,3, y=0,1,2, y < or = x c f_X(x) f_Y(y) mu_X sigma_X^2. mu_Y sigma_Y^2 Cov?(X,Y) rho

A certain raw material is classified as to moisture content \(X\) (in percent) and impurity \(X\) (in percent). Let \(X\) and \(Y\) have the joint pmf given by (a) Find the marginal pmfs, the means, and the variances. (b) Find the covariance and the correlation coefficient of \(X\) and \(Y\). (c) If additional heating is needed with high moisture content and additional filtering with high impurity such that the additional cost is given by the function \(C=2 X+10 Y^{2}\) in dollars, find \(E(C)\). Equation Transcription: Text Transcription: X Y C=2X+10 Y^2 E(C)

Let the random variables X and Y have the joint pmf f(x, y) = x + y 32 , x = 1, 2, y = 1, 2, 3, 4. Find the means X and Y , the variances 2 X and 2 Y , and the correlation coefficient .

Let X and Y have the joint pmf defined by f(0, 0) = f(1, 2) = 0.2, f(0, 1) = f(1, 1) = 0.3. (a) Depict the points and corresponding probabilities on a graph. (b) Give the marginal pmfs in the “margins.” (c) Compute \(\mu_{X}, \mu_{Y}, \sigma_{X}^{2}, \sigma_{Y}^{2}, \operatorname{Cov}(X, Y)\), and \(\rho\). (d) Find the equation of the least squares regression line and draw it on your graph. Does the line make sense to you intuitively?

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 X , Y , 2 X , 2 Y , Cov(X, Y), and . (b) Find the equation of the least squares regression line and draw it on your graph. Does the line make sense to you intuitively?

Let X and Y have a trinomial distribution with parameters n = 3, pX = 1/6, and pY = 1/2. Find (a) E(X). (b) E(Y). (c) Var(X). (d) Var(Y). (e) Cov(X, Y). (f) . Note that =  pX pY /[(1 pX )(1 pY )] in this case. (Indeed, the formula holds in general for the trinomial distribution; see Example 4.3-3.)

Let X and Y be random variables with respective means \(\mu_X\) and \(\mu_Y\), respective variances \(\sigma^2_X\) and \(\sigma^2_Y\), and correlation coefficient \(\rho\). Fit the line y = a + bx by the method of least squares to the probability distribution by minimizing the expectation \(K(a, b) = E[(Y ? a ? bX)^2]\) with respect to a and b. Hint: Consider \(\partial K/\partial a = 0\) and \(\partial K/\partial b = 0\), and solve simultaneously.

The joint pmf of X and Y is \(f(x, y) = 1/6, 0 \le x + y \le 2\), where x and y are nonnegative integers. (a) Sketch the support of X and Y. (b) Record the marginal pmfs \(f_X(x)\) and \(f_Y(y)\) in the “margins.” (c) Compute Cov(X, Y). (d) Determine \(\rho\), the correlation coefficient. (e) Find the best-fitting line and draw it on your figure.

Let the joint pmf of X and Y be \(f(x, y) = 1/4,\quad (x, y)~ \epsilon ~S = {(0, 0), (1, 1), (1, -11), (2, 0)}\). (a) Are X and Y independent? (b) Calculate Cov(X, Y) and \(\rho\). This exercise also illustrates the fact that dependent random variables can have a correlation coefficient of zero.

A certain raw material is classified as to moisture content X (in percent) and impurity Y (in percent). Let X and Y have the joint pmf given by (a) Find the marginal pmfs, the means, and the variances. (b) Find the covariance and the correlation coefficient of X and Y. (c) If additional heating is needed with high moisture content and additional filtering with high impurity such that the additional cost is given by the function \(C = 2X + 10Y^2\) in dollars, find E(C).

A car dealer sells X cars each day and always tries to sell an extended warranty on each of these cars. (In our opinion, most of these warranties are not good deals.) Let Y be the number of extended warranties sold; then Y X. The joint pmf of X and Y is given by f(x, y) = c(x + 1)(4 x)(y + 1)(3 y), x = 0, 1, 2, 3, y = 0, 1, 2, with y x. (a) Find the value of c. (b) Sketch the support of X and Y. (c) Record the marginal pmfs fX (x) and fY (y) in the margins. (d) Are X and Y independent? (e) Compute X and 2 X . (f) Compute Y and 2 Y .(g) Compute Cov(X, Y). (h) Determine , the correlation coefficient. (i) Find the best-fitting line and draw it on your figure.

If the correlation coefficient exists, show that satisfies the inequality 1 1. Hint: Consider the discriminant of the nonnegative quadratic function that is given by h(v) = E{[(X X ) + v(Y Y )]2}.