Let X = (X1, X2, ..., Xn)T be a random vector with mean vector μ and covariance matrix Σ. Suppose that for every a = (a1, a2, ..., an)T ∈ Rn, the random variable aTX has a (one-dimensional) Gaussian distribution on R. a) For fixed a ∈ Rn, compute the moment generating function MaTX(u) of the random variable aTX, writing the answer using μ & Σ. b) Define MX(t1, t2, ... , tn), the moment generating function of the random vector X, and express it in terms of MtTX, the moment generating function of tTX where t = (t1, t2, ..., tn). c) Combining a) and b), compute the moment generating function MX(t) of X and hence prove that the random vector X has a Gaussian distribution.

Let X = (X1, X2, ..., Xn)T be a random vector with mean vector μ and covariance matrix Σ. Suppose that for every a = (a1, a2, ..., an)T ∈ Rn, the random variable aTX has a (one-dimensional) Gaussian distribution on R. a) For fixed a ∈ Rn, compute the moment generating function MaTX(u) of the random variable aTX, writing the answer using μ & Σ. b) Define MX(t1, t2, ... , tn), the moment generating function of the random vector X, and express it in terms of MtTX, the moment generating function of tTX where t = (t1, t2, ..., tn). c) Combining a) and b), compute the moment generating function MX(t) of X and hence prove that the random vector X has a Gaussian distribution.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 24EQ

See similar textbooks

Similar questions

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 the
Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxy
Let X be a discrete random variable with support {0,2,3}, and suppose E(X) =1 and P(X=0)=P(X=2). Find Var(X)
Assume that the random variables X1 and X2 are bivariate normally distributed with mean µ and covariance matrix Σ. Show that if cov(X1, X2) = 0 then X1 and X2 are independent.
Suppose that Z₁, Z2, ..., Zn are statistically independent random variables. Define Y as the sum of squares of these random variables: n Y=)Z (n>2) i=1 (a) Express the moment generating function My(t) of the random variable Y in terms of moment generating functions involving the random variables Z, i = 1, ..., .., n. (b) Determine My(t) for the special case that Z₁ ~ N(0, 1). (c) For the above special case, calculate E[Y] by using the moment generating function.
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))
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 ).
Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
Let X, Y and Z be three independent random variables such that E(X)=E(Y)=E(Z)=1 and E(X2)-E(Y²)=2E(Z²)=10. Then Var(X+Y+Z) is equal to:
Let (Ω, Pr) be a probability space, and let X and Y be two independent random variables that are positive and have non-zero variance. (a) Prove that X^2 and Y are independent. Note that by symmetry, it will also follow that Y^2 and X are independent. (b) Use the result from part (a) to show that the random variables W = X + Y and Z = XY are positively correlated (i.e. Cov(W, Z) > 0).
Suppose Y1, Y2, Y3, and Y4 are mutually independent and identically distributed exponential random variables with mean 1. a) Find the moment generating function of the sum T = Y+Y2+Y3+Y4. What is the distribution of T? b) Find the pdf of the maximum order statistic Y(4).
Let B be an n-dimensional random vector of zero mean and positive- definite covariance matrix Q. Suppose measurements of the form y = WB are made where the rank of W is m. If B is the linear minimum variance estimate of B based on y, show that the covariance of the error B - B has rank n - m.