Skeleton Answers
Midterm, CSC 354S, Erindale Version
Spring 2000

1. [binomial]
The RV X that denotes the # of successes in n Bernoulli trials, with constant probability p of success, has a binomial distribution. X can be considered to be the sum of n independent Bernouuli RVs, each with mean p and variance p(1-p) = pq; X = X1 + X2 + ... Xn.
The mean is given by
	E(X) = p + p + ... + p = np
and the variance by
	Var(X) = pq + pq + ... + pq = npq = np(1-p).

2. [bus]
The passenger has to wait more than 5 minutes only if the arrival time is between 7:00 and 7:15 or between 7:20 and 7:30. If X is a RV that denotes the # of minues past 7:00 that the passenger arrives, the desired probability is P(0 < X < 15) + P(20 < X < 30). X is a uniform RV on (0,30). Therefore the desired probability is given by
F(15) + F(30) - F(20) = 15/30 + 1 - 20/30 = 5/6.

3. [MLE]
L(b) = 1/(b**n) and we must have 0 <= Xi <= b for all i. Thus, L(b) is maximized by setting b to be as small as possible, i.e., as close to 0 as possible as 0 < b. This gives
b hat = X sub (n), the last order statistic or the largest Xi.

4. I've mentioned this in class many times.

5. deals with topics not covered in lectures yet.