Let X ~ Uniform(0.05) (uniform random variable) with Y = (X + 2)-2. Use the delta...
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] = ax +b and Var[Y|X =x] = 1 a) compute E[Y] b) compute Var[Y] c) Find E[XY]
Let X be a uniform(0, 1) random variable and let Y be uniform(1,2) with X and Y being independent. Let U = X/Y and V = X. (a) Find the joint distribution of U and V . (b) Find the marginal distributions of U.
2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-? 2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-?
2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-? 2. Let Xbe a random variable with a continuous uniform density between -1 and 1, i.e, X U(-1,1) Random variable Y is defined by the following transformation: (1) Var(Y)-?
Let the random variable X have a uniform distribution on [0,1] and the random variable Y (independent of X) have a uniform distribution on [0,2]. Find P[XY<1].
Q6 (4pt) Let X be a discrete uniform random variable over {1,2,...,6} and let Y be a Bernoulli random variable with parameter 1/2 such that X, Y are independent. (1) Find the PMF of the random variable Z, where Z XY. (2) Compute the third moment of Z, that is, E[z2
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
b) et X be uniform [O, 1] and let Y be an independent random variable uniform on [O, 2]. Find the density of W = log(X) and identi fy the distrib