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 EX? and EX”. 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

soluation :- a) Let & be the fixed point Hence X~POB (A) P (x-x) = 1=0, 1,2.. th 2 a = Mean e=2.71821 We know that E(X)=2=1= Mean V(x) = 0=1= Variance = 62 Hence by given E(X)= 1 Var (4)= E(X) - [[(x)]" E(xi- var (x) + [E(x)? - lt2=2 E(x²)=2 But E(X)=n then E(X3)=3 (b) PEIX-1123] = S(x-11 23 frasda SS1X-1123 1X- 11 fraud = {1x-11 23 (x-19 fruch < 1 2 $ (x-13² fraude P[18-1123] = 1 *1 = 11 q

C) P (31000 3500 | 55002255) = P (51000 500) P (ssoo >255) -(500) = eno a(255) = e (500-255) en (245) But 2-1 -245 P (51000 25001 S5oo Z255)

NOTE:: I hope your happy with my answer......**Please support me with your rating

**Please give me"LIKE".....Its very important for me......THANK YOU