11. 3 points Suppose we have ten independent random variables Xi~N1,1), X2N (21) X1N(10,1) Find the variance of the product of the random variables. In other words, find 10 11. 3 points Suppose...
Say we have data xi, . . ,z,, which are independent and identically distributed normal random variables with mean μ and variance 100. How often does this interval cover 11, 20
Say we have data xi, . . ,z,, which are independent and identically distributed normal random variables with mean μ and variance 100. How often does this interval cover 11, 20
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
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant a. Name it. a= b) According to the Borel-Cantelli Lemmas, does a.s /n In other words, will there eventually reach a point in the sequence of random variables where every X a?
Suppose Xi, X2,.. are independent Geometric (number of trials) random variables where ~Geometric p 1- a) It is easily shown that Xfor some constant...
Suppose X1, X2,... are independent Geometric (number of trials)
random variables where Xi ~ Geometric(p = 1/i^2)
a) It is easily shown that Xn converges to a for some constant
a. Name it.
b) According to the Borel-Cantelli Lemmas, does Xn almost surely
converge to a?
Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
Properties of Expectation and Variance Suppose we have two independent discrete random variables, say X1 and X2. Suppose further E(X1) = 21 Var(X1) = 126 and E(X2) = 3.36 Var(X2) = 1.38 Compute the Expectations and Variances of the following linear combinations of X1 and X2. a) E(πX1 + eX2 + 17) b) E(X1 · 3X2) c) Var( (√ 13X2) + 46) d) Var(X1 + 2X2 + 14)
3. Suppose that X1, X2, , Xn are independent random variables with the same expectation μ and the same variance σ2. Let X--ΣΑι Xi. Find the expectation and variance of
(20 pts) Suppose that Xi,X2 X3, X4 are independent random variables, all of which have mean and variance ơ2 Compute the expected value and variance of the following (a) Y1 = 0.25X1 + 0.25X2 + 0.25X3 + 0.25X4 (b) ½ = 0.1X1 + 0.2X2 +0.3X3 + 0.4X4 (c) Y3 = 0.5X1 + 0.4X2 + 0.3X3-0.2X4 What do you observe about the expectation of Yi, Y2, Ys? Which of these random variables has the LEAST variance?
Let X1 and X2 be independent random variables with mean μ and variance σ2. Suppose we have two estimators 1 (1) Are both estimators unbiased estimatros for θ? (2) Which is a better estimator?
[PLEASE USE HINT]
Problem 10: 10 points Suppose that (Xi, X2,... are independent identically distributed binary variables taking (0 1) values with probability PX here 0<q Introduce the new variable, M such that the event {M = k} occurs when three consecutive successes appear at the first time. In other words, event [M = kj, where k-3, occurs if and only if and there is no previous occurrence of three consecutive successes. Use conditioning techniques to derive the expected value,...
Suppose we have 5 independent and identically distributed random variables Xi,X2.X3,X4,X5 each with the moment generating function 212 Let the random variable Y be defined as Y -XX. The density function of Y is (a) Poisson with λ-40 (b) Gamma with α-10 and λ-8 (c) Normal with μ-40 and σ-3.162 (d) Exponential with λ = 50 (e) Normal with μ-50 and σ2-15