4. We are given two independent random variables X & Y that define a new one W=X+Y. The first RV Yhas the moment generating function (MGF) þy(s) = 1/(1 - s). The second RV X has the momentgenerating function x(s) = 1/(1 — s) 4a. What are E[Y], E [Y²] and E[Y³]b. What is E[W²]Hint: Consider the examples in Chp 9 on sum of two variable and the Table given for the MGFs.

We have given that the random variables & are independent.& the moment generating function…

Answered: 4. We are given two independent random…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis- tribution. Suppose that E[X₁] = E[X₂] µ₁, that E[Y₁] = E[Y₂] μ2 and that var(X₁) var (X₂) = var (Y₁) = var(Y₂) = o² > 0. Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via S² = 2 Σ(Xi − X)² + Σ(Y₂ − Ỹ)² ΣΥ i-n²) i=1 1 (2₁x i=1 = (1/√2 1/√2 0 The aim of this question is to describe the joint distribution of X - Y and S² (a) Let U be a 4 4 matrix whose first two rows are 0 0 1/√2 1/√₂) - Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use plenty of zeros!] (b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁ and V₂. Use this to describe the joint distribution of X - Y and S².
1.1 Consider Y₁, Y2, ‚Yn are i.i.d. random variables with n > 4 and T is the sample mean Y. Also let W = 1 Y¿. Then (A) E(Y₁ +Y₂|W) = E(Y₁+Y₂) (B) E(W|T) = E(Y₁|T) (C) E(Y₁+W|W) = E(Y3 + Y4|W) (D) none of the above.
2.5.7. Let the random variables X1 and X2 have the joint pdf f(x1,x2) = 1/T, for (x1 – 1)2 + (2 +2)2 < 1, zero elsewhere. Find fi(æ1) and f2(x2). Are X1 and X2 independent? %3D
2. Suppose that Y1 ~ Exp(3) and Y2 ~ Exp(2) are independent. (a) Determine the joint pdf of Y1 and Y2. (b) Use the Jacobian method to determine the pdf of U = Y1 + Y2. (c) Use the mgf method to determine the distribution of V = 4Y1 + 6Y2. The distribution of V will be one of the "famous" ones in your Super Handout.
6. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.
Find E(R) and V (R) for a random variable R whose moment-generating function ismR(t) = e2t(1-3t2)-1
Suppose that X and Y are random variables with E(X) = 1, E(X²) = 5, E(Y) = 3, and E(XY) = 1. Match up the following quantities. Cov(X,Y) Corr(X, Y) Var(-X/2) Cov(X/2,-2Y + 3) Drag answer here Cannot be determined Drag answer here Drag answer here 2 Cannot be determined 1
A pseudo-random signal (S) of -3 or 3 is transmitted over a noisy channel. S=-3 is transmitted with probability of 1/3. The received signal, Y, is the sum of the sent signal and the noise, N. The noise is modeled as a normal Gaussian with E[N]=0,var(N)=4 . The received signal is decoded as -3 for Y≤−0.5 and 3 for Y>−0.5. a. What is the probability of error in this system? b. To improve the system, each signal is transmitted three times and output is determine by majority rule (2 or more). What is the new probability of error?
Which of the following is false for continuous and independent random variables Y and Z?
8. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).
4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?
Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx(x) = 2e-2x;x 2 0 and fy (y) = 2e-2y;y > 0 Define a new random variable Z = X + Y. Derive moment generating function of Z, i.e. Mz(t).

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