Question

Problem 5. (5 pts) Expectation Trick, Strong Law and the CLT (a) (2 pts) Let X be the number of fixed points in a random permutation of n elements. We know that EX = 1. Compute EX2 and EX3. Hint : Write X as a sum of indicator random variables. The number of fixed points in a random permutation is also known as the matching problem. (b) (1pt) For the same X as in part (a), give Markov and Chebyshev bounds on the probability: P(|X – 11 > 3). = (c) (2pts) Let Sn U1 + U2 + + Un be a sum of independent uniform random variables on [0, 1]. Approximate the probability: P(S1000 > 500|S500 > 255)

Answer 1

0 Solution : Z z Z z Given that @ back x be the number of fived pornlo. we know that E(X)=1 mean važiance - 02-V(X)= 1-1 P(x=0) ¿?? ad X=0, 1, 2, 7=mcan=1 e=207184 Hence by given E(X)=) Vatcx) = E(X2)-[E(X)* Elx?) - Vau(x) + +[Ecx] - 1+0102 2 1+1=2 EĽx2)-2 But E (+2) = 1 Then / E(X)=3. 6 112-11-3 Flolda P[10-1/23). Spa-123 sf4-1123 (-1123 124=14 Fe06) do esla-1123 (0-12 scanda sý Soca-v* scando 2 a ܝ

P[]Y-1] 23 ] - + P[ 13-1] 33] - 4 © P(51000 2006 | హుం 205), 3 P[5000 వరం) PLgu2255) 7 (500) C-1/255) A //100-200) - Ad45) ful ): 1 -245 -245 P( 102 100 1 00002 205) - 2

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