homework5 - Homework 5 36-325/725 due Friday Oct 5 (1) Let...

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Unformatted text preview: Homework 5 36-325/725 due Friday Oct 5 (1) Let . Hint 1: first find the cdf of . and let . Hint 2: (2) A particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump the probability is that the particle will jump one unit to the left and the probability is that the particle will jump one unit to the right. Let be the position of the particle after units. Find and . (This is known as a random walk.) (3) A fair coin is tossed until a head is obtained. What is the expected number of tosses that will be required? be a continuous random variable with cdf and suppose that and that exists. Show that . Hint: Consider integrating by parts. The following fact is helpful: if exists then . (6) Let be Plot versus for (Recall that the Cauchy has density such a difference. 1 0' ' 0 4 1 ' 1 QkQ"u"r ~ }| z6 x m als{`ynenem ((( (00 4 $$lw2$vgV V u P htP i sqdp rV 4 10 uv(' o UnlG m random variables and let . . Repeat for . .) Explain why there is 1 1g "ED ' $&1 'E Ag wD 4 1 ' 1 ' g" ED (5) Let 4 ' x1 3 wD (4) Let be a discrete random variable. Let . By definition, . In class, we learned the lazy statistician rule: . Prove that . HIFQ P H F IG 3 B @ 86 % 4 C2A975 3 if and only if 1 ' a" ED 1 ' r1 ' y 4 1 ' tr H "u&"e Gi vHQusqp i 1 ' g" e 4 3 V X X `0 Y V0 3 WUTS4 R 1 ' $ ED 3 10 ' %# 2)(&$"! ( 4 j1 k"' ih0g ey % Y f d 1"use wi ' r1 ' y 1vH'Qursqpi h1 3 ED t H 4 ' 0IQ$" 7 4 1( ' 1 'ec $" fdb . Find for (8) A simulated stock market. Let be independent random variables such that . Let . Think of as "the stock price increased by one dollar", as "the stock price decreased by one dollar" and as the value of the stock on day . (8a) Find and . (Hint: see question 2). (8b) Simulate way to do this is: and plot versus for . An easy n <- 10000 y <- rbinom(n,1,.5) y <- 2*y-1 ### do you see what this does? x <- cumsum(y)/(1:n) help(cumsum) Repeat the whole simulation several times. Notice two things. First, it's easy to "see" patterns in the sequence even though it is random. Second, you will find that the four runs look very different even though they were generated the same way. How do the calculations in (8a) explain the second observation? The moral of the story: when you lose lots of money in the stock market one day, remember that I warned you. 2 0 4 sP 3 ((( (0 0 4 $$lw2ewl$V 1 'ec $" fdb $" ED 1 ' V n 0 Y 4 dP 3 P 3 htP i 4 0 4 10 Y 4 ' 4 10 4 ' $uv`udvsP 3 EI`uvP 3 7 m 3 3 V 3 (7) Let and let . Find and 1 3 ab 'ec 1 3 wD ' 4 10 uv(' o . ...
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