1. A bag contains 6 coins, 2 of which have a head on both sides while the other 4 coins are normal. A coin is chosen at random from the bag and tossed 2 times. The number of heads obtained is a random variable, say X.

(a) What are the possible values of X? Also tabulate the pmf of X.

(b) Calculate E(X) and Var(X).

2. The moment-generating function of a random variable X is given by M(t) = 0.3e t + 0.4e 2t + 0.2e 3t + 0.1e 5t .

(a) Find the pmf of X.

(b) Find the values of µ and σ 2 for X.

(c) Calculate P(X ≥ 2).

(d) Calculate E(2X).

3. Police are to conduct random breath testing on drivers on a busy road one Friday evening. Suppose 3% of the drivers drink and drive at the time. Let X be the number of drivers that police need to test to get the first case of drinking and driving. Let Y be the number of drivers tested to get 3 such cases.

(a) Name the probability distribution and specify the value of any parameter(s) for each of the two random variables X and Y .

(b) What is the probability that at least 4 drivers are to be tested to get the first drinking and driving case?

(c) What is the probability that exactly 30 drivers are to be tested to get 3 drinking and driving cases?

(d) On average, how many drivers do police need to test to get 3 cases of drinking and driving?

(e) Find P(Y > 50).

(f) Suppose the police have not found any case of drinking and driving in their first 5 tests. What is the conditional probability that the police still cannot find any case in their next 5 tests?

(g) What is the probability that the police will find exactly 4 cases of drinking and driving in testing 200 drivers? At least two different probability distributions can be used to calculate/approximate this probability. Which two, and how different are the results?

4. A jar contains 5 green jelly beans and 5 purple jelly beans.

(a) Suppose one jelly bean is to be selected at random. What is the value of p, the probability that a purple jelly bean is selected? 

(b) Suppose 9 jelly beans are to be selected at random with replacement. Let X be the number of purple jelly beans in the selected. Calculate P(X = 1), the probability that there is exactly one purple jelly bean in the 9 selected.

(c) Now suppose 5 beans are to be selected at random without replacement. Let Y be the number of purple jelly beans in the selected. Calculate P(Y = 1), the probability that there is only one purple jelly bean in the 5 selected.

5. The moment-generating function of a random variable X is given by M(t) = e 3(e t−1) .

(a) Give the name of the distribution of X (if it has a name).

(b) Find the values of µ and σ 2 for X. (Note: No need to derive µ and σ 2 if you know the name of the distribution.)

(c) Calculate P(1 ≤ X ≤ 2).

6. The moment-generating function of a random variable X is given by M(t) = 0.25e t /1 − 0.75e t , t < − ln(0.75).

(a) Give the name of the distribution of X (if it has a name).

(b) Find the values of µ and σ 2 for X. (Note: No need to derive µ and σ 2 if you know the name of the distribution.)

(c) Calculate P(X ≥ 2).