5. Suppose that X1, X2, and X3 are independent random variables such that E[Xl i =...
5. Suppose that Xi, X2, and X3 are independent random variables such that EX i12,3. Find the value of ElX(2X1- X3)21. dEX? 1 for
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
1. Suppose that X1, X2, and X3 E(X1) = 0, E(X2) = 1, E(X3) = 1, Var(X1) = 1, Var(X2) = 2, Var(X3) = 3, Cov(X1, X2) = -1, Cov(X2, X3) = 1, where X1 and X3 are independent. a.) Find the covariance cov(X1 + X2, X1 - X3). b.) Define U = 2X1 - X2 + X3. Find the mean and variance of U.
6. Suppose random variables X1, X2, X3 have the following properties: E(X1) = 1; E(X2) = 2; E(X3) = −1 V(X1) = 1; V(X2) = 3; V(X3) = 5 COV (X1,X2) = 7; COV (X1,X3) = −4; COV (X2,X3) = 2 Let U = X1 −2X2 + X3 and W = 3X1 + X2. (a) Find V(U) (b) Find COV (U,W).
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,...
Consider the independent random variables X1, X2, and X3 with - E(X1)=1, Var(X1)=4 - E(X2)=2, SD(X2)=3 - E(X3)=−1, SD(X3)=5 (a) Calculate E(5X1+2). (b) Calculate E(3X1−2X2+X3). (c) Calculate Var(5X1−2X2).
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).
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. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3. 1. What is the distubution for each Yį, i = 1, 2, 3? 2. What is the expected value for Y1 + Y2 +Yz? 3. Are Yį and Y2 independent? Explain it by computing their joint P.M.F. 4. What is the variance of Y1...