5. Suppose that Xi, X2, and X3 are independent random variables such that EX i12,3. Find...
5. Suppose that X1, X2, and X3 are independent random variables such that E[Xl i = 12,3. Find the value of E(X 2x1-X3)2]. mt random variables such that EX-0 and ER-i for 0 and ELX 1 for
5. [22] If Xi, X2, and X3 are independent random variables with E(XI) 4, E(X2)-3, E(X3)2, V(X) = 1, V(X:) = 5, V(Xs) = 2, and Y = 2X1 + X2-3X1 (a) Determine E(Y) and V(Y). P(Y > 2.0) and P( 1.3 Y 8.3).
5. Suppose that three random variables Xi, X2, and X3 have a continuous joint distribution with the following p.d.f. (x1+2x2+3z3) and f(1, r2, 3) 0 otherwise. (a) Determine the value of the constant c; (b) Find the marginal joint p.d.f. of Xi and X3; (c) Find P(Xi < 1|X2-2, X3-1)
If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1): If X1, X2, and X3 are three independent Uniform random variables (Xi-Unif(0,1)) a) Use the convolution integral to find density function of Z-x1+X2+X3. b) What is E[Z]? independent Uniform random variables (Xi-Unifo,1):
3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3. 3. Suppose that X1, X2, X3 be i.i.d. random variables with P(Xi 0) 2/5 and P(X 1) 3/5. Find the MGFof X, + X2 + X 3.
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
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...
5. Suppose that X X', and X are independent random variables such that ELX.]-O and BLX?) -1 for is 12.3. Find the value of EIX/(2x1-X3M
Suppose we have 5 independent and identically distributed random variables X1, X2, X3, X4,X5 each with the moment generating function 212 Let the random variable Y be defined as Y = Σ Find the joint probability that all Xi, (i-1,.5), are larger than 9.
Given three random variables Given three random variables Xi, X2, and X such that X[Xi X2 Xa, 2 1 0.5 1 (a) EX, + c) var(X2- X3 (d) var(X2 + X3) (e) cov(4X2 +X1,3Xi - 2X3)