2. Let U and V be independent random variables, with P(U 1) 1/4 and P(U =...
Problem 4. Let X and Y be independent Geom(p) random variables. Let V - min(X, Y) and Find the joint mass function of (V, W) and show that V and W are independent
Problem 3: 10 points σ2. Define Assume that U, V, and W are independent random variables with the same common variance X= + W and Y-V-W. 1. Find the variances Var[X] and Var[Y 2. Find the covariance between X and Y, that is: cov [x,Y 3. Find the covariance between (X+Y) and (X - Y), that is: COV[(X +Y), (X -Y)]
There are two independent Bernoulli random variables, U and V , both with probability of success 1/2. Let X=U+V and Y =|U−V|. 1) Calculate the covariance of X and Y 2) Explain whether X and Y are independent or not 3) Identify the random variable expressed as the conditional expectation of Y given X, i.e., E[Y |X].
4.2 The Correlation Coefficient 1. Let the random variables X and Y have the joint PMF of the form x + y , x= 1,2, y = 1,2,3. p(x,y) = 21 They satisfy 11 12 Mx = 16 of = 12 of = 212 2 My = 27 Find the covariance Cov(X,Y) and the correlation coefficient p. Are X and Y independent or dependent?
9. Let X and Y be independent and identically distributed random variables with mean u and variance o. Find the following: (a) E[(x + 2)] (b) Var(3x + 4) (c) E[(X-Y)] (d) Cov{(X + Y), (X - Y)}
2. Suppose U and V are independent geometric random variables with parameter p. Let Z-U+V. Determine the conditional probability mass function of pujz(-In) of U given that Z- n
Solve #2, #3, and #4 1) Let X, Y be random variables with joint probability density function Find k and P(X 3, Y s 2). 2) For X,Y as in problem 1) find the marginal densitiesx(a) of X and fr (y) of Y. Are X, Y independent? Explain your answer. 3) For X, Y as in problem 1) find Cov(X, Y). 4) For X, Y as in problem 1) find the conditional probability
2. Let X and Y be independent, standard normal random variables. Find the joint pdf of U = 2X +Y and V = X-Y. Determine if U and V are independent. Justify.
Let X and Y be independent exponential random variables with parameter 1. Find the joint PDF of U and V. U = X + Y and V = X/(X + Y)
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y...