a)
The probability distribution table with marginal distribution of X and Y is,
Y | X = -1 | X = 0 | X = 1 | X = 2 | P(Y) |
-2 | 1/5 | 0 | 0 | 0 | 1/5 |
-1 | 0 | 1/5 | 0 | 0 | 1/5 |
1 | 1/5 | 0 | 0 | 0 | 1/5 |
2 | 0 | 0 | 1/5 | 0 | 1/5 |
3 | 0 | 0 | 0 | 1/5 | 1/5 |
P(X) | 2/5 | 1/5 | 1/5 | 1/5 | 1 |
(b)
E(XY) =
= 2 * 3 * (1/5) + 1 * 2 * (1/5) + -1 * 1 * (1/5) + 0 * (-1) * (1/5) + (-1 ) * (-2) * (1/5)
= 1.8
E(X) = -1 * (2/5) + 1 * (1/5) + 2 * (1/5) = 0.2
E(Y) = -2 * (1/5) - 1 * (1/5) + 1 * (1/5) + 2 * (1/5) + 3 * (1/5) = 0.6
Cov(X, Y) = E(XY) - E(X) E(Y) = 1.8 - 0.2 * 0.6 = 1.68
E(X^2) = (-1)^2 * (2/5) + 1^2 * (1/5) + 2^2 * (1/5) = 1.4
SD(X) = = 1.16619
E(Y) = (-2)^2 * (1/5) + (-1)^2 * (1/5) + 1^2 * (1/5) + 2^2 * (1/5) + 3^2 * (1/5) = 3.8
SD(Y) = = 1.854724
Corr(X,Y) = Cov(X,Y) / (SD(X) * SD(Y)) = 1.68 / (1.16619 * 1.854724) = 0.7767132
c)
The conditional variable of Y | X is,
P(Y = -2 | X = -1) = P(X = -1, Y = -2) / P(X = -1) = (1/5) / (2/5) = 1/2
P(Y = 1 | X = -1) = P(X = -1, Y = 1) / P(X = -1) = (1/5) / (2/5) = 1/2
P(Y = -1 | X = 0) = P(X = 0, Y = -1) / P(X = 0) = (1/5) / (1/5) = 1
P(Y = 2 | X = 1) = P(X = 1, Y = 2) / P(X = 1) = (1/5) / (1/5) = 1
P(Y = 3 | X = 2) = P(X = 2, Y = 3) / P(X = 3) = (1/5) / (1/5) = 1
d)
E(Y | X) =
E(Y | X = -1) = -2 * (1/2) + 1 * (1/2) = -1/2
E(Y | X = 0) = -1 * 1 = -1
E(Y | X = 1) = 2 * 1 = 2
E(Y | X = 2) = 3 * 1 = 3
Let the frequency function of the joint distribution of the random variables X and Y P(X...
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
3. Let the random variables X and Y have the joint probability density function 0 y 1, 0 x < y fxy(x, y)y otherwise (a) Compute the joint expectation E(XY) (b) Compute the marginal expectations E(X) and E (Y) (c) Compute the covariance Cov(X, Y)
3. Let the random variables X and Y have the joint probability density function fxr (x, y) = 0 <y<1, 0<xsy otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
1. Let the joint probability (mass) function of X and Y be given by the following: Value of X -1 -1 3/8 1/8 Value of Y1 1/8 3/8 (a) Determine the marginal (b) Determine the conditional distribution of X given Y (c) Are they independent? d) Compute E(X), Var(X), E(Y) and Var(Y). (e) Compute PXY <0) and Ptmax(X,Y) > 0 (f) Compute Elmax(X, Y)] and E(XY) (g) Compute Cov(X,Y) and Corr(X, Y) 1
Question 4: (5 Marks) Let X and Y be continuous random variables have a joint probability density function of the form: f(x,y) = cy2 + x 0 SX S1, 0 Sys1. Determine the following: 1. The value of c. 2. The marginal distributions f(x) and f(y). 3. The conditional distribution f(xly). 4. Are X and Y independent? Why? - the
Question 4: Let X and Y be two discrete random variables with the following joint probability distribution (mass) function Pxy(x, y): a) Complete the following probability table: Y 2 f(x)=P(X=x) 1 3 4 0 0 0.08 0.06 0.05 0.02 0.07 0.08 0.06 0.12 0.05 0.03 0.06 0.05 0.04 0.03 0.01 0.02 0.03 0.04 2 3 foy)=P(Y=y) 0.03 b) What is P(X s 2 and YS 3)? c) Find the marginal probability distribution (mass) function of X; [f(x)]. d) Find the...
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
you have two random variables, X and Y with joint distribution given by the following table: Y=0 | .4 .2 4+.26. So, for example, the probability that Y 0, X - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),f(r). (b) Find the conditional distribution (pmf) of Y give X, denoted f(Y|X). (c) Find the expected values of X and Y, E(X), E(Y). (d) Find the variances of X...
55. Let X and Y be jointly continuous random variables with joint density function fx.y(x,y) be-3y -a < x < 2a, 0) < y < 00, otherwise. Assume that E[XY] = 1/6. (a) Find a and b such that fx,y is a valid joint pdf. You may want to use the fact that du = 1. u 6. и е (b) Find the conditional pdf of X given Y = y where 0 <y < . (c) Find Cov(X,Y). (d)...
Let X and Y be continuous random variables with joint distribution function: f(x,y) = { ** 0 <y < x <1 otherwise What is the P(X+Y < 1)?