28. (a) Under what conditions do we say that two random variables X and Y areindependent?(b) Demonstrate that if X and Y are independent, then it follows that E(XY) =E(X)E(Y);(e) Show by a counter example that the converse of (ii) is not necessarily true.

Answered: 28. (a) Under what conditions do we say…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 2EQ: 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed...

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2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1
3. Let the random variable X have the pmf f(x) = a) E(X) b) E(X²) c) E(3X²2X + 4) (x+1)² 9 for x = -1,0,1. Compute
ii) Let X be a random variable taking three values: P(X= a₁) = P₁, P(X=a₂) = P2, P(X=a3) = P3, where p₁ + P2 + P3 = 1 and P₁, P2, P3 € (0, 1). Let A = {X = a₁} and G = {N, 0, A, Ac}. Prove that E (X³|G) = a³¹₁ + ª²P² + a²P3, P2 + P3 1A.
2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
4. (a) Consider a nonnegative, integer valued random variable Y , and a random sample X1,., Xn. + Xy) in terms of E(X1) Assume Y and all the X;'s are independent. Find E(X1+ and E(Y). ... (b) Suppose each person who logs onto Amazon.com on Christmas Eve is expected to spend $80. One hundred people are randomly chosen to see how much is spent by them. Each of them visits Amazon.com with probability 60%. How much money do we expect this group of people to spend at Amazon.com?
(a) Let X and Y be independent random variables both with the same 0. Define a new random variable W aX+by, where a and mean b are constants. (i) = Obtain an expression for E(W). What constraint is there on the values of a and b so that W is an unbiased estimator of u? Hence write all unbiased versions of W as a formula involving a, X and Y only (and not b).
3. If X and Y are independent uniform (0, 1) random variables, show that 2 (a + 1)(a + 2) E[|X - Yº] = for a>0
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Let X1, X2, ..., X, be independent random variables and Y = min{X1, X2, ..., Xm}. Fy (y) = 1 – || (1 – Fx,(y)) i=1 (a) A certain electronic device uses 5 batteries, with each battery to have a life that is exponentially distributed with mean of 48 hours and is independent of the life of other batteries. If the device fails as soon as at least one of its batteries fail, what is the expected life of the device?
For each of the identities in (a)-(f), choose an appropriate description from the following three options. Briefly explain your answers. (i) this identity holds in general (for any random variables X and Y). |(ii) this identity does not hold in general, but it holds if X and Y are independent. (iii) this identity may not hold even if X and Y are independent. (a) Var(X + Y) = Var(X)+ Var(Y) (b) E(XY) = (EX)(EY) (c) E(X + Y) = EX+EY (d) Var(XY) = Var(X)· Var(Y). (e) E[(X – EX)Y] = 0 (Hint: first distribute the multiplication by Y and use the linear property of the expectation, keeping in mind that EX is a constant.)
The moment generating function of the random variable X is given by mx(s) = e2e-2 and the moment generating function of the random variable Y is my (s) = (et + )10. If it is assumed that the random variables X and Y are independent, find the following: (a) E(XY); (b) E[(X – Y)²] (c) Var(2X – 3Y).

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28. (a) Under what conditions do we say that two random variables X and Y are independent? (b) Demonstrate that if X and Y are independent, then it follows that E(XY) = E(X)E(Y); (e) Show by a counter example that the converse of (ii) is not necessarily true.