1. Consider the following distribution of (X,Y) where X and Y are both binary random variables:...
1. Consider the following distribution of (X,Y) where X and Y are both binary random variables: JA if (z, y) = (0,0) 1/8 3/8 if(x, y) = (1,0) if (x, y) = (0, 1) y(2, y) = 1/4 if(x, y) = (1,1). () What is the covariance between Y and X?
1. Consider the following distribution of (X Y) where X and Y ae both binary random variables: 1/4 i (a)-(0.0 1/4 if (x, y) (0,0) 1/8 if (r,y) (1,0) Jx3/8 if (r,)- (0,1) ,Y (z, y) = 1/4 if (, ) (11 (a) What is the probability density function of Y? (b) What is the expectation of Y1 (c) What is the variance of Y? (d) What is the standard deviation of Y? (e) Do the same to X. (f)...
1. Assume that X and Y are both binary random variables. Assume that there are constants, Bo and B such that Y-Ao + AX + t. Assume Elu 1 X-0. (a) Express EY | X-0nters of Bo and B1. (b) Express EY | X = 1] in terms of A, and A. (c)Assume that the joint pdf for (X, Y) is 1/4 f(,y) (0,0) 1/8 if(x, y) = (1,0) 3/8 if (r, y) (0,1) 1/4 if (x,y) - (1, 1)....
1. Let X and Y have a discrete joint distribution with ( P(X = x, Y = y) = {1, 10, if (x, y) = (-1,1) if x = y = 0 elsewhere Show that X and Y are uncorrelated but not independent. [5 points] 2. Let X and Y have a discrete joint distribution with f(-1,0) = 0, f(-1,1) = 1/4, f(0,0) = 1/6, f(0, 1) = 0, $(1,0) = 1/12, f(1,1) = 1/2. Show that (a) the two...
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter = 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(x > 0.25) U (Y > 0.25)}? (c) What is the conditional distribution of X. given that Y - 3? (d) What is Var(Y - E[2X] + 3)? (e) What is...
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
14. Random variables X and Y have a density function f(x, y). Find the indicated expected value. f(x, y) = (xy + y2) 0<x< 1,0 <y<1 0 Elsewhere {$(wyty E(x2y) = 15. The means, standard deviations, and covariance for random variables X, Y. and Z are given below. LIX = 3. HY = 5. Az = 7 Ox= 1, = 3, oz = 4 cov(X,Y) = 1, cov (X, Z) = 3, and cov (Y,Z) = -3 T = X-2...
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter A= 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(X > 0.25) U (Y> 0.25)}? nd (c) What is the conditional distribution of X, given that Y =3? ur worl mple with oumbers vour nal to complet the ovaluato all...
Suppose the joint probability distribution of two binary random variables X and Y are given as follows. x/y 1 2 0 3/10 0 1 4/10 3/10 X goes along side as 0 and 1, Y goes along top as 1 and 2. a) Show the marginal distribution of X. b) Find entropy H(Y ). c) Find conditional entropy H(X|Y ) and H(Y |X). d) Find mutual information I(X; Y ). e) Find joint entropy H(X, Y ). f) Suppose X...
ka (3) [6 pts] X and Y are discrete random variables with the following joint distribution: 14 22 30 065 102 0.05 0.10 0.03 0.01 Value ofX 50.17 0.15 0.05 0.02 0.01 8 0.02 0.03 0.15 0.10 (a) Calculate the probability distribution, mean, and variance of Y (b) Calculate the prohability distribution, mea, and variane of Y given X (c) Calculate the covariance and correlation between X and Y 8