13. Let X₁, X2,... be i.i.d. r.v.'s with finite expectation and finite variance o², and let XL²→μ as n →∞.n1/1 X₁. Show that Xni=1=

13. Given that X1,X2,... be i.i.d. r.v.'s with finite expectation μ and finite variance σ2 . So,…

Answered: 13. Let X₁, X2,... be i.i.d. r.v.'s…

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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10
1. Let X ~ Poisson(A) and Y ~ Poisson(u). Assume that X and Y are independent. Use probability generating functions to find the distribu- tion of X + Y.
11. Suppose X and Y are independent continuous random variables uniformly dis- tributed on the intervals 0
3 Let X - exp (1). We define Y = [X] +1 and Z = X - [X], where [t] stands for the integer part of t (e.g., [T] = 3). 1. Find the distribution of Y. 2. Compute the expectancy and variance of Y. 3. Compute the CDF of Z.
Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o2 is unknown. Consider the estimate n 1 Σ(X₁-1 for the variance. a) Show that 7² C := i=1 -μ)² 72 2 ·Σ( X ₁ = μ)² ~ x² (n). O i=1 b) For a € (0, 1), denote the a-quantile of x2 (n) by qn(a). Using this notation, find numbers an, bn E R such that P(o² bno²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o against the alternative H₁: 02 08. d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis Ho: o² = 4 vs. H₁: o² #4 at significance level 5%.
3. Let X Geometric(p) with pe (0, 1). Prove that for every a, beN we have P(X = a+ b|X > a) = P(X = b). 4. Let X be a discrete random variable with the following distribution: P(X = -3) = a, P(X = -2) = ; P(X = 0) = 6: P(X = -1) = b, %3| P(X = 1) = c, P(X = 2) = 32 a. Find a, b, c. b. Calculate E(3+ 2X), E(3+2x)², Var(3+ 2X). 5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p).
Let S. be the time it takes a standard Brownian motion to hit a. Find the probabilities 1) P(S2 < S_4), 1) P(S < S_2 < S4), 2) P(S1 < S_2 < S1 < S12).
21. Let X1, X2, ... be independent, U (0, 1)-distributed random variables, and let Nm E Po(m) be independent of X1, X2, Set V= max{X1,..., (Vm = 0 when Nm =0). Determine (a) the distribution function of Vm, (b) the moment generating function of Vm. It is reasonable to believe that V is "close" to 1 when m is "large" (cf. Problem 8.1). The purpose of parts (c) and (d) is to show how this can be made more precise. (c) Show that E V, (d) Show that m(1 - V) converges im distribution as m → x. and de- termine the limit distribution. →l as m→.
be i.i.d. random variables with expectation 1 and finite 34. Let X1, X, variance o, and set S, = X1+ X2 +-+X,, for n21. Show that VS,-Vn N (0,6*) as and determine the constant 6. and positive, finite

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13. Let X₁, X2,... be i.i.d. r.v.'s with finite expectation and finite variance o², and let X = L² E1 X₂. Show that X →μas n→∞. 1 n