Let ?1~?(5,15) and ?2~?2(15), then calculate P(X1-5>3√X2). (chi-sqr(15)) .independent
Let ?1~?(5,15) and ?2~?2(15), then calculate P(X1-5>3√X2). (chi-sqr(15)) .independent
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...
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 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
Let the independent random variables X1 and X2 have binomial distribution with parameters n1 = 3, p =2/3, and n2=4, p=1/2 respectively. Compute P(X1 = X2).Hint: List the four mutually exclusive ways that X1 = X2 and compute the probability of each.
Let X1 and X2 be independent rolls of a fair four sided die. Compute P(X1 ≥ 2*X2|X1 ≤ X2^2 ).
Solve b
3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, respectively. (a) Show that Yı = X1/X2 and ½ = X1 + X2 are independent and that ½ is x 2) (b) Deduce that and KJr2 (X1+X2)/(n+r2) are independent F-variables.
3.6.14. Let Xi, X2, and Xs be three independent chi-square variables with rı, r2, and T3 degrees of freedom, respectively. (a) Show that Yı = X1/X2 and ½ =...
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...
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1 = 2X1 and Y2 = 2X2. Then A E[Y1 · Y2] = 2/9 BE[Y1 · Y2] = 4/9 C P[Y1 · Y2 = 0) = 1/9 D P[Y1 · Y2 = 0) = 2/9 (3) Let X and Y be two independent random variables having gaussian (normal) distribution with mean 0 and variance equal 2. Then: A P[X +Y > 2] > 0.5 B...
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).
(a) Let x, have a chi-squared distribution with parameter y, and let x, be independent of x, and have a chi-squared distribution with parameter vz Show that x2 + x, has a chi-squared distribution with parameter v, + vz Let y = x2 + x2. Identify the correct expression for Fyn). "1/2 - 2x212/2-12-(x1 + x2)/2 dx1 OFyly) = ' -X1/2 dx1 + -x212 dxz e 109) = 6 {1*(***) ** (*)*" + x2)120mg +6 (277(3)) -<*****@*)*** .(-)70%as C (7-(1)...