5. Let X be a non-central χ%,A) random variable, and Y, independent of X, be a...
5. Let X be a non-central x2 (5, A) random variable, and Y, independent of X, be a x(4) randonm variable. (a). Derive the moment generating function of 2X - 1, and find its mean and variance. (b). Find the mean of W =
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].
5. (a) (6 marks) Let X be a random variable following N(2.4). Let Y be a random variable following N(1.8). Assume X and Y are independent. Let W-min(x.Y). Find P(W 3) (b) (8 marks) The continuous random variables X and Y have the following joint probability density function: 4x 0, otherwise Find the joint probability density function of U and V where U-X+Y and -ky Also draw the support of the joint probability density function of Uand V (o (5...
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
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
I. Suppose that χ ~ Poisson (2) and y ~ Poisson (3) are independent random variables. (a) Find the probability generating function of χ + y. (b) Use part (a) to find P(χ + y = 13). 2. Suppose that χ ~ Poisson (2) and y ~ Geom(0.25) are independent random variables. (a) Find the probability generating function of . (b) Find the probability generating function of χ + y.
Assume X is a normal random variable with mean 20 and variance 16, and Y is a Gamma random variable with parameters 5 and 2. In addition X and Y are independent. Construct a box with length L = [X], width W = 2|X|, and height H = Y. Let V be the volume of the box. Calculate the expected value E[V]?
Let a random variable X be uniformly distributed between −1 and 2. Let another random variable Y be normally distributed with mean −8 and standard deviation 3. Also, let V = 22+X and W = 13+X −2Y . (a) Is X discrete or continuous? Draw and explain. (b) Is Y discrete or continuous? Draw and explain. (c) Find the following probabilities. (i) The probability that X is less than 2. (ii) P(X > 0) (iii) P(Y > −11) (iv) P...
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
Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y Q4) Let X and Y be two independent N(0,1) random variable and 10 ei Find the covariance of Z and W.WE3-Y