Problem 9. let X1, X2, ,Xn be independent 0,1) random variables. Set Is there a matrix...
Problem 9. Let Xi, X2,... , Xn be independent 2/ (0,1) random variables. Set F(t) Is there a matrix M such that holds with independent standard normal random variables Z1, Z2, Z3? If so, calculate M.
Problem 8. Let X1, X2, , Xn be independent ฆ(0,1) random variables. Let m,-1 for k 1,2,3. Are there numbers mi,m2, m3 such that n.y rn1 m1 a.S n3 m3 holds? If so, calculate these numbers.
The independent random variables X1, X2, ... Xn are each uniformly distributed on (0,1). M is the minimum number of X's that sum to a value of at least one. (so if X1 = .4, X2, = .5, and X3 = .3, M would be 3 since 3 X values were needed for the sum of all the X's to be at least 1). a. What is the probability mass function of M. b. What is the expected value of...
Let X1, X2, ....,. Xn, be a set of independent random variables, each distributed as a normal random variable with parameters μί and σ. Let х, ai Use properties of moment generating functions to determine the distribution of Y, meaning: find the type of distribution we get, and its expected value and variance
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
Problem 10. Let X1, X2, . . . be random variables such that Xn → c in D holds for some number c. Show that Xn → c in P holds
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn) uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (X1, X2,..., Xn) and N -min (X1, X2,..., Xn) 1. Find the joint distribution of a pair (M,N) 2. Derive the CDF and density for M 3. Derive the CDF and density for N.
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W) (5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
Let X1 and X2 be independent n(0,1) random variables. Find the pdf of (X1 - X2)^2/2