A symmetric random walk takes place on the integers 0, 1, 2, ..., N with absorbing barriers at 0 and N, starting at k. Show that the probability that the walk is never absorbed is zero.

Q1. a) In a group of 80 students, 50 have a laptop (A), 30 have a tablet (B) and 3 have both a laptop and a tablet. i) Draw a Venn diagram to represent this information. A 47 3 27 B 3 ii) Find the probability that a student either only has a tablet or does not have a laptop or both, using the equation P(A' U B) = P(A') + P(B) – P(A' ʼn B).

Suppose that X;; i = 1,2, 3, .. n are independent random variables with a common uniform distribution over (0, 1) and Yg is distributed over (0, Yg -1) with Y, =1,k = 1,2, ..., n. Let Z = k = 1,2,3, .n R.D.R EGALA STAT 203 March 4, 2021 Show by mathematical induction induction or any other method that Zg has probability density function (-In z)*-1 fR(z) = k = 1,2,3, . n (k – 1)! Hence deduce that Zg and Y, have the same distribution (k = 1,2,3, ..n)

A person’s birthday occurs on a day i with probability P i , where i = 1, ..., n. (Of course, P 1 + ... + P 2 = 1.) Assume independent assignment of birthdays among different people. In a room with k people, let P k = P k (p 1 , ..., p n ) be the probability that no two persons share a birthday. Show that this probability is maximized when all birthdays are equally likely: p i = 1/n ∀i.