Homework 07

This homework follows the standard late penalty: 0% if in the stat310 mailbox by Thursday 15 Mar 4pm, 10% by 5pm the following day, 100% otherwise. Please read the syllabus for other homework policies.

  1. (8 pts) Let \(X \sim Exp(1) \) and \(Y \sim Exp(1) \). X and Y are independent.

    1. Find the pdf of \(A = \frac{X + Y}{2} \) and \(B = \frac{X - Y}{2} \).
    2. Are A and B independent?
  2. (8 pts) \(f(x,y) = \theta e^{-(x+\theta y)}, x>0, y>0, \theta>0 \).

    a. Find the pdf of A = X * Y.

    b. Is this a named distribution?

  3. (8 pts) Let \(R \sim Unif(0,1) \) and \(A \sim Unif(0,2\pi) \). R and A are independent.

    1. Find the pdf of \(X = R \cos(A) \) and \(Y = R \sin(A) \). (Hint: think polar!)
  4. (5 pts) Suppose you are taking an online quiz that has 10 multiple choice questions that has 4 options. You did not study at all for the quiz, so you’re just going to guess at the answer.

    1. (1 pt) What is the probability that you pass (that is, you get at least 6 questions right)?

    2. (3 pts) If you are allowed to take it twice and only take the highest score, what is the probability that you pass? What if you are allowed 3 times?

    3. (1 pt) How many times do you have to re-take the quiz to have 90% chance of passing?

  5. (3 bonus pts) Let \(X_{1}, X_{2}, X_{3}, … X_{n} \) be independent, with the same CDF, \(F \).

    1. Find the pdf of \(Y = min(X_{1}, X_{2}, X_{3}, … X_{n}) \)