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- Let X ~ N(0, 2) and Y ~ covariances Cov(X,Y) and Cov(Y, X +Y). Exp(A = 3) be two uncorrelated random variables. Find theLet X be a Gaussian random vector with distribution X-N (BR) ~ 5 Let Y be a Gaussian random vector formed by multiplying X by a certain matrix: Find the mean vector of Y. Y₂ = X₁ X₂Let (Z8, +8) and H={0,2,4,6} .Find the normalizer of (H, +8) in (Z8, +8).
- Let X be an n-dimensional random vector and the random vector Y be defined as Y = AX + b, where A is a fixed m by n matrix and b is a fixed m-dimensional vector. Show that CY = ACXAT .Show that if X and Y are independent Exp(a)-distributed random vari- ables, then X/Y E F(2,2).Let X1,X2, .,Xn be i.i.d. random variables, where X; ~ Bernoulli(p). Define Y1 = X1X2, Y2 = X2X3, Yn-1 = Xn-1Xn, Y, = X,X1. If Y = Y1 + Y2 +……+ Yn, find 1. E[Y], 2. Var(Y).
- Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).Compute the norm of the vector 7 = |||Let X and Y be independent and N(0, 1) distributed random variables. Let U = X and V = X/Y. Show that the random variable V is Cauchy distributed and find E(V ).
- TRUE OR FALSE. a.) Let X (X₁, X2,..., Xn)' be a random vector with joint cumulative distribution function 2 - fx(-). Then (Xi − #i)² has a chi-square distribution with n degrees of freedom where 72 i=1 Hand o are the mean and variance, respectively, of each of the random variable in X, i = 1,2,..., n. b.) Let X₁, X2,..., Xn be a random sample from a density f(), which has mean and finite n 0² n variance o2, and let X = n X₁. Then E[X] = µ and Var(X) = μ = c.) For any random variables X and Y, E[XY] = E[X] E[Y].Let Z₁ and Z₂ be independent standard normal random variables. Let pe [-1, 1]. Find a matrix L such that has a distribution. X = LZ N (1). (1))What is the dimension of the subspace of the space of 5 x 5 matrices (aij) sucn that a11 +...+ a55 = 0?
