Let X be a normal random variable with mean 0 and variance 0.5 and Y be exponentially distributed with mean 1. Suppose X and Y are independent. Find P(Y>X2 ).
Let X be a normal random variable with mean 0 and variance 0.5 and Y be...
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
. Suppose that Y is a normal random variable with mean
µ = 3 and variance σ
2 = 1; i.e.,
Y
dist = N(3, 1). Also suppose that X is a binomial random variable
with n = 2 and p = 1/4; i.e.,
X
dist = Bin(2, 1/4). Suppose X and Y are independent random
variables. Find the expected
value of Y
X. Hint: Consider conditioning on the events {X = j} for j = 0, 1,
2.
8....
Let X variable Y by be a normal random variable with mean 0 and variance 1. We define the random y2 if x 20, Y= (a For t E R, compute Mr()-Elen'], the moment generating function of Y. Compute EY
Let X1 be a normal random variable with mean 2 and variance 3, and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. What is the distribution of the linear combination Y = 2X1 + 3X2?
Let X be a random variable with mean μ and variance σ2, and let Y be a random variable with mean θ and variance τ2, and assume X and Y are independent. (a) Determine an expression for Corr(X Y , Y − X ). (b) Under what conditions on the means and variances of X and Y will Corr(XY, Y −X) be positive (i.e., > 0 )?
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y is the sum of independent random variables, compute both the mean and variance of Y. (b) Find the moment generating function of Y and use it to compute the mean and variance of Y.
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y...
Let X be a normal random variable with mean 0 and variance σ^2. Find the density for |X|.
X is a random variable exponentially distributed with mean Y, where Y is uniformly distributed on the interval [0,2], Find P(X>2|Y>1) roblems:
X is a random variable exponentially distributed with mean Y, where Y is uniformly distributed on the interval [0,2], Find P(X>2|Y>1) roblems: