Question

For each question on a multiple-choice test, there are five possible answers of which exactly one is correct for each question. Assume there are 10 questions on the test and a student selects one answer for each question at random. Let X be the number of correct answers he or she gets.

a) How is X distributed? (Specify the values of the corresponding parameters).

b) Find P(X < 6) and P(X = 6). c) Find E(X) and V ar(X).

2. Suppose that a basketball player makes a free throw 60% of the time. Let X equal the number of free throws that this player must attempt to make a total of 3 shots. Assume independence among attempts,

a) How is X distributed? (Specify the values of the corresponding parameters)

b) Find P(X = 6).

c) Find E(X) and V ar(X). 2 3.

A factory puts biscuits into boxes of 100. The probability that a biscuit is broken is 0.03. Find the probability that a box contains 2 broken biscuits using Poisson approximation.

4. (20 pts) Let X have the p.d.f: f(x) = 3x 2 2 for −1 ≤ x ≤ 1 and f(x) = 0 for otherwise . a) Find P(X > 0.5). b) Find E(X). 3 c) Find the c.d.f F(x). d) Find π0.5. 4

5. (10 pts) Suppose that a system contains a certain type of component whose time, in years, to failure is given by X. The random variable X is modeled nicely by the exponential distribution with mean time to failure θ = 3. Find: a) P(X ≥ 4). b) Given that the component has been in operation for 2 years, find the conditional probability that it will last for at least another 4 years. 5

6. (8 pts) If X ∼ N(−10, 25), find: a) P(X ≤ 0) b) P(−15 ≤ X ≤ −5) c) P(X ≤ 20) d) π0.4.

7. (6 pts) If Z ∼ N(0, 1), find the values of z such that: a) P(Z > z) = 0.3. b) P(|Z| ≤ z) = 0.4. 6

8. (21 pts) Jim sells blueberry bushes to his customers when they are at least 18 inches tall. He wants to know how long it will take each of his blueberry bushes to grow tall enough to sell. To get an estimate of this time, he selects ten plants at random and records the number of days each one takes to grow from a seed into an 18 inch tall plant. 96, 98, 97, 101, 98, 95, 100, 95, 98, 101. Find the values of the following statistics a) Sample mean ¯x b) Sample median ˜x. c) Sample mode and sample range. d) Sample variance and sample standard deviation.

Answer 1