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Problem 5. (5 pts) Expectation Trick, Strong Law and the CLT (a) (2 pts) Let X...
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...
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 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...
= (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)
Let f [n]n] be a permutation. A fixed point of f is an element x e [n] such that f(x)-x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X 2? (c) What is the probability that X--1? (d) What is the expectation of X? (Hint: As usual, express X as...
(1) Let f : [n] [n] be a permutation. A fixed point of f is an element x e [n] such that f(x) - x. Now consider random permutations of [n] and let X be the random variable which represents the number of fixed points of a given permutation. (a) What is the probability that X 0? (b) What is the probability that X-n -2? (c) What is the probability that X-n-1? (d) What is the expectation of X? (Hint:...
3. Let U1, U2,. be a sequence of independent Ber(p) random variables. Define Xo 0 and Xn+1-Xn +2Un-1, 1,2,.. (a) Show that X, n 0,1,2, is a Markov chain, and give its transition graph. (b) Find EX and Var(X) c)Give P(X
8. (15 points) Let X ~ Binomial(30,0.6). (a) (5 points) Using the Central Limit Theorem (CLT), approximate the probability that P(X > 20). (b) (5 points) Using CLT, approximate the probability that P(X = 18). (c) (5 points) Calculate P(X = 18) exactly and compare to part(b).
O RANDOM VARIABLES AND DISTRIBUTIONS Expectation and variance of a random variable Let X be a random variable with the following probability distribution: Value x of X P(X-) 0.35 0.40 0.10 0.15 10 0 10 20 Find the expectation E (X) and variance Var(X) of X. (If necessary, consult a list of formulas.) Var(x) -
I. (5 points) Let X be a random variable with moment generating function M(t) = E [etx]. For t > 0 and a 〉 0, prove that and consequently, P(X > a inf etaM(t). t>0 These bounds are known as Chernoff's bounds. (Hint: Define Z etX and use Markov inequality.)
Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of...