## Probability distribution

1. You go to the supermarket with your friend. After picking up something, each of you arrives at a different queue at the same time. Assume that the time you need to wait(measured in minutes) has a geometric distribution with mean 2 and the waiting time of your friend is also geometrically distributed but with mean 4.
a) Find the pmf of the difference between the waiting times of you and your friend.
b) Find the probability that you wait longer than your friend.

2. Suppose that 40% of voters are in favor of certain legislation. A large number of voters are polled and a relative frequency estimate for the above proportion is obtained. Use the Chebyshev inequality to determine how many voters should be polled in order to make sure that the probability that the relative frequency estimate differs from the actual probability by
less than 0.01 is at least 0.95.

3. Let the number of widgets tested in an assembly line in 1 hour be a binomial random variable with parameter n=600 and p . Suppose that the probability that a widget is faulty is q. Denote S as the number of widgets that are found faulty in a 1-hour period. Find the mean and variance of S.