You have 1000 dollars to put in an account with interest rate R, compounded annually. That is, if X, is the value of the account at year n, then X, = 1000(1+ R)", for n = 0,1, 2, -... The value of Ris a random variable that is determined when you put the money in the bank, but it does not not change after that. In particular, assume that R~ Uniform(0.04, 0.05). a. Find all possible sample functions for the random process {Xn,n = 0, 1,2, ...}. b. Find the expected value of your account at year three. That is, find E[X3].

4. In professor Shankar's quiz, the time taken to complete the quiz is a random variable with a pdf £x (x) = 20 = exp(-26 ) ₁ x 20² 20 1₂x ≥0 What is the average time for completion? If there are 50 students in the class, what is the probability that at least 20 of them will complete the quiz in less than the average? What is the probability that exactly 35 students complete within the average time.

X ~ Exp (1/9.848) 1) Find the expected value 2) Find the standard deviation

A coin which has probability 0.19 of coming up Heads is tossed three times. Let Y be the number of Tails observed. Find E(Y^2), giving your answer correct to two decimal places.