hw3_3225 - P ( Y 5). Then, nd its ex-act value using...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Homework 3, Math 3225, Fall 2010 October 24, 2010 1. Find an example of a pair of random variables X and Y such that E ( XY ) n = E ( X ) E ( Y ). 2. Compute (and provide proof) of the moment generating function for the discrete random variable having mass function p ( x ) = p x + r 1 r 1 P p x (1 p ) r , where x = 0 , 1 , 2 , ... , and where r 1 is some integer constant. (For the other values of x , we set p ( x ) = 0.) 3. Suppose we have a population of individuals living on a remote island. Let X denote the number of individuals living there at some time t , and then let Y be the number of individuals living there at time t + 1. We will assume here that X and Y are random variables – there is some uncertainty associated to their values. Let us suppose that X has a binomial distribution with parameters n = 1 , 000 and p = 1 / 2. Further, suppose that P ( Y = y | X = x ) = b 1 / 2 , if y = 2 x ; 1 / 4 , if y = 3 x ; 1 / 4 , if y = x. Compute E ( Y ) by using the tower property of expectation. 4. Suppose that Y = ( X 1 + ··· + X 100 ) / 100, where X 1 , ..., X 100 are inde- pendent Poisson random variables each having parameter λ = 5. Using the Central Limit theorem, approximate
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: P ( Y 5). Then, nd its ex-act value using properties of Poisson random variables. How do the answers compare? 1 5. Suppose that X = X 1 + + X k , where the X i s are all independent and have the same distribution. Find a formula for E ( X 4 ) in terms of the moments 1 = E ( X 1 ), 2 = E ( X 2 1 ) , 3 = E ( X 3 1 ) and 4 = E ( X 4 1 ). (Hint: This is almost a triviality if you use moment generating functions.) 6. Let T denote the triangle with corners at (0 , 0) , (0 , 1), and (1 , 1), and dene the function f ( x, y ) = b x + cx 2 y 2 , if ( x, y ) T ; , otherwise . Find the constant c that makes this function into a probability density function. 7. Suppose X and Y are i.i.d. random variables having pdf f ( t ) = b 1000 /t 2 , if t > 1000; , otherwise . Suppose Z = X/Y . Determine the pdf for Z . 2...
View Full Document

This note was uploaded on 10/23/2011 for the course MATH 3225 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

Page1 / 2

hw3_3225 - P ( Y 5). Then, nd its ex-act value using...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online