For each question on a multiple-choice test, there are fivepossible answers of which exactly one is correct for each question.Assume there are 10 questions on the test and a student selects oneanswer for each question at random. Let X be the number of correctanswers he or she gets.
a) How is X distributed? (Specify the values of thecorresponding 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% ofthe time. Let X equal the number of free throws that this playermust attempt to make a total of 3 shots. Assume independence amongattempts,
a) How is X distributed? (Specify the values of thecorresponding 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 thata biscuit is broken is 0.03. Find the probability that a boxcontains 2 broken biscuits using Poisson approximation.
4. (20 pts) Let X have the p.d.f: f(x) = 3x 2 2 for −1 ≤ x ≤ 1and f(x) = 0 for otherwise . a) Find P(X > 0.5). b) Find E(X). 3c) Find the c.d.f F(x). d) Find π0.5. 4
5. (10 pts) Suppose that a system contains a certain type ofcomponent whose time, in years, to failure is given by X. Therandom variable X is modeled nicely by the exponential distributionwith mean time to failure θ = 3. Find: a) P(X ≥ 4). b) Given thatthe component has been in operation for 2 years, find theconditional probability that it will last for at least another 4years. 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 whenthey are at least 18 inches tall. He wants to know how long it willtake each of his blueberry bushes to grow tall enough to sell. Toget an estimate of this time, he selects ten plants at random andrecords the number of days each one takes to grow from a seed intoan 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) Samplevariance and sample standard deviation.
