Assume X is a normal random variable with mean 20 and variance 16, and Y is a Gamma random variable with parameters 5 a...
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 and Y be independent normal random variables with parameters E[X] =ux, E[Y] = uy and Var(X) = x, Var(Y) = Oy. Indicate whether each of the following statements is true or false. Notation: fx,y (x, y), fx(x), fy (v) denote the joint and marginal PDFs of X and Y , respectively; $(x) is the CDF of a standard normal random variable with zero mean and unit variance. E[XY]=0
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 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.
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 )?
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what value of a is the variance of W a minimum? σ: Let W-aX + (1-a) Y, where 0 < a < 1. Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random...
1. Suppose X ∼ Gamma(a,b) and Y ∼ Gamma(c,d). Furthermore suppose X and Y are independent. Let W = X + Y . (a) Find the MGF of W. (b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?