Homework 4 Due Wednesday, October 13th, 2021 1. Compute the following expectations (a) Let X ∼ Normal(3, 0.1), compute E[2X − 1] (b) Let X = 7! k=1 k · Yk where Yi ∼ Uniform(0, 3) are independent. Compute Var(X) (c) Let X ∼ Binomial(2, 15 ) and Y = X3, compute E[Y ] 2. Let X be a discrete random variable with values {0, 1, . . . , n}. Show: E[X] = n−1! k=0 P (X > k) 3. A factory manager verifies the state of the machines in the factory. The probability of having a failure within the first 5 years of operation is 30%. Among the machines which had a failure in the first five years, the prob- ability of having a more significant failure subsequently and permanently going out of order is 75%. Among the machines which did not have a failure in the first 5 years, the probability of going permanently out of order is 40%. (a) What is the probability for a machine to go permanently out of order? (b) What is the probability that a machine which is permanently out of order did not have a failure in the first 5 years? (c) Select 10 machines at random, each older than 5 years. Let X be the number of machines which had a failure within the first 5 years. Write the pmf of X. (d) Find E[X] and Var(X) 4. Let X be a standard normal, i.e. X ∼ Normal(0, 1) and let Y = eX . Using the rule of the lazy statistician, E[Y ] = E[eX ] = 1√ 2π " R exe− 1 2x 2 dx = 1√ 2π " R e− 1 2 (x 2−2x)dx = 1√ 2π " R e− 1 2 ((x−1)2−1)dx = e 1 2 1√ 2π " R e− 1 2 (x−1)2dx = e 1 2 Where we have used rules of exponents, completing the square, and the fact that shifting the pdf φ horizontally doesn’t change that it integrates to 1. We have shown E[Y ] = √ e, now compute Var(Y ). 1 5. A particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump the probability is p that the particle will move one unit to the left and the probability is 1− p that the particle will jump one unit to the right. Jumps are independent of one another. Ultimately, we are going to be concerned with the position of the particle after n jumps. (a) Let Yi be the change in position after jump number i. What is the pmf of Yi? Hint: You may want to write Yi as a function of a Zi ∼ Bernoulli(p) (b) Write Xn in terms of the Yi (or Zi if you prefer) (c) Compute E[Xn] (d) Compute Var(Xn) 2