Section Number Points Comments/Hints
6.6 2 3
4 3
9(a) 4 Hint: use Chebyshev's Inequality
12 6
19(b) 3
6.7 6 4
8(c) 3
11 5 Warning: this is one of the harder problems
6.8 7 3 DAM2, 6.6, 7
Extra problems:
1. [5 points] A fair die is rolled twice. Let X denote the sum of the
outcomes and
Y the first outcome minus the second.
(a) Are X and Y independent?
(b) What is Cov(X,Y)? [Hint: While you can compute this directly by
enumerating all outcomes, an easier way to do this is to write X=X1+X2
and Y=X1-X2, where X1 is the outcome of the first throw and X2 is the
outcome of the second throw.]
2. [5 points] You are trying to estimate the probability that a coin lands heads. How many times do you have to toss the coin to guarantee that your esimate is within 0.01 of the true probability with probability at least .9. That is, if p is the true probability, you toss the coin n times, and #H is the number of heads you get, you want Pr(|#H/n -p| ≤ 0.01) ≥ 0.9. (Hint: use Chebyshev's inequality.)
3. [3 points] From past experience, a professor know that the test score of a student taking the final exam is a random variable with mean 75. Use Markov's inequality to get an upper bound for the probability that a student's test score is greater than 85.
4. [3 points] Suppose that in addition the professor knows that the
variance of the test score is 25. What is the probability that a
students scores between 65 and 85.