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Unformatted text preview: Homework 6 36325/725 due Friday Oct 12 (1) Suppose we generate a random variable X in the following way. First we flip a fair coin. If the coin is heads, take X to have a Unif(0,1) distribution. If the coin is tails, take X to have a Unif(3,4) distribution. (a) Find the mean of X. (b) Find the standard deviation of X. (2) Let and be constants. Show that be random variables and let (3) Let Find . (4) Let r(x) be a function of x and let s(y) be a function of y. Show that Also, show that . (5) Prove that Hint: Let 1 P and let Bear in mind that . Note that is a function of . Now write 4 D lD B B 0 0 D @ 1 0 BCA8970 ( & 6(&' 45320 0 ( & )('& #! " ' ' ' ' $% D B X gAt r w utr qi vyxvs2p8 ` T 4 R GF g hb R Y` fdcaD R YWPB V USD 2 QPB I HAE b e b P b ` X i nD B mD B 4 4 D jP 4 B kjQPB D B 4 i ! 4 D B e X B e 4 B D hgt fD t 5SD t D shdD AD B B 4 B B B 4 B B D AD B D 5SD #D D B and Expand the square and take the expectation. You then have to take the expectation of three terms. In each case, use the rule of the iterated expectation: i.e. You may want to use the result of problem (4) to evaluate one of the terms. (7) This question is to help you understand the idea of a sampling distribution. be iid with mean and variance . Let . Then is a statistic, that is, a function of the data. Since is a random variable, it has a distribution. This distribution is called the sampling distribution of the statistic. (7a) Find the mean and variance of the sampling distribution, i.e. find and . (7b) Don't confuse the distribution of the data and the distribution of the statistic . To make this clear, here is an example. Let . Let be the density of the . Plot . Now let . Find and . Plot them as a function of . Comment. Now we will simulate the sampling distribution. nsim < 10000 ### think of nsim as essentially being infinity n < 25 xbar < rep(0,nsim) for(i in 1:nsim){ x < runif(n,0,1) xbar[i] < mean(x) } hist(xbar) expected.value < mean(xbar) variance < var(xbar) print(expected.value) 2 D B  F E h A D At B D#e `nQ8w B t F AE D B z Let (6) Show that if uncorrelated. D i D u ti D vutvyB B 4pvyB qD B XD B 4 q 4 B rvD i D B svD B B 9rD i B pD o
for some constant then ' } s ~h4 6(' x q{ x 4 ywD R 4 B and are F AE '  } 4 D B t 6(' #e g `nvQ8w h dF E D t B print(variance) Repeat this experiment for . Check that the simulated values of and agree with your theoretical calculations. What do you notice about the sampling distribution of as increases?  D t B `` ` e 4 Ae ` s g Ae s g 
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This note was uploaded on 05/25/2008 for the course STAT 36625 taught by Professor Larrywasserman during the Fall '01 term at Carnegie Mellon.
 Fall '01
 LarryWasserman

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