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 M...
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)
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
how to calculate cov(x1,x2), cov(x2,x3),cov(x3,x1)? and how to calculate var(x1),var(x2),var(x3)? Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 E [X] ,1-10 | and var(X)=Σ-| 0 3 0. 1 0.5 1 compuite: 2
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...
Let X1, X2, X3 … be independent random variable with P(Xi = 1) = p = 1-P(Xi=0), i ≥ 1. Define: N1 = min {n: X1+…+ Xn =5}, N2 = 3 if X1 = 0, 5 if X1 = 1. N3 = 3 if X4 = 0, 2 if X4 = 1. Which of the Ni are stopping times for the sequence X1, …?
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0. Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
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
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...
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):