Problems4.3. Suppose that X and Y are jointly discrete random variables withfor x = 1, 2, .n and2p(x, y) = {n(n + 1)y = 1, 2, .otherwiseCompute Px(x) and py(y) and determine whether X and Y are independent.

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Answered: Problems 4.3. Suppose that X and Y are…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Example 10-3. Show that if X', Y’ are the deviations of the random variables X andY from their respective means, then 1 (1) r =1- 2N X'; Y;\ Ox Oy,
5. Let {S, n>0} be a simple random walk with So = 0 and S₁ = X₁ + ... + X, for n ≥ 1, where X₁, i = 1,2,... are independent random variables with P(X; = 1) = p, P(X; = -1) = q = 1 -p for i>1. Assume p‡q. Put Fo= {2,0}, Fn = 0(X₁, X2, ..., Xn), n ≥ 1. Let b, a be two fixed positive integers. Define T = min{n: S₁ = -a or S₁₁=b}. Assume P(T<∞) = 1. Explain why one can use Doob's Optional Stopping Theorem to conclude that E[ZT] = 1. We can use the conclusion that: T is a stopping time with respect to the o-fields Fn, n ≥ 0. A) B) Define Z₁ = (q/p) Sn, n ≥ 0. {Zn, n ≥ 0} is a martingale with respect to the o-fields Fn, n ≥ 0.
4. Continuous random variables X and Y are independent, with parameters ux = 30, ly = 40, Ox = 15, and ơy = 5. Which of the following gives the correct values for ux+Y and ox+y? = 70 and ox+y = 14.1 (A) Hx+Y (B) Мx+ү %3D 70 and o x+y %3D 15.8 (C) Hx+Y = 70 and ox+y = 20 (D) Ux+Y = -10 and ox+Y =14.1 (E) Ux+y =-10 and ox+y = 15.8
Let X be a discrete random variable with the following PMF 1 1 for x=-2 8 100 for x = -1 Px(x) = for x = 0 1 for x = 1 for x = 2 4 0 otherwise a) Find E[X] and Var(X) b) Now we define a new random variableY = (x+1)², Find PMF of Y and E[Y]
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13-17. If a random variable X assumes the values 0 and 1 with P(X=0) = 3 P(X= 1), then nX) is : () 16 (ii) is 3 (iii) 16 (iv) 4. i6 m n t m 2/6
2. Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X1, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞ Hint: to lower bound nSn > x, if suffices to demonstrate that there is an empty interval of some length x/n which contains none of the points. Make a list of some candidate intervals and then use Cheby- shev's inequality to show one of them is empty with probability going to 1.
4. A particle moves on the x- axis starting at x = 0. With equal probability, the particle moves to the left or right one unit. Let X denote the position on the x- axis after four moves. (a) Find px(x). Notice that X can take on -4,-2,0, 2, 4. Fill out the table below. xPx(x) (b) What is the expected position of the particle after four moves. That is, compute E(X). Express your answer as a fraction in lowest terms.
7. Suppose that the joint p.d.f. of two random variables X and Y is as follows: (4 – 2x – y) for x > 0, y > 0, f(x, y) = and 2x + y 2|X = 0.5).
(L&K 4.3) Suppose that X and Y are jointly discrete random variables with 2 p(x, y) = { n(n+1) for x = 1, 2, ..., n and y = 1, 2, ..., x, 0 otherwise. Compute the pmfs px (x) and py (y) and determine whether X and Y are independent.

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Problems 4.3. Suppose that X and Y are jointly discrete random variables with for x = 1, 2, .. y = 1, 2, otherwise n and 2 p(x, y) = {n(n + 1) Compute Px(x) and py(y) and determine whether X and Y are independent.