For each random variable: • State the distribution that will best model random variable. Choose from the common distributions:

(1) Uniform,
(2) Exponential, or
(3) Normal distribution.

Explain your reasoning. State the parameter values that describe the distribution. Give the probability density function.

a. Random Variable 1: In the board game Perfection, individuals must attempt to fit a number of shaped plastic pieces into their matching holes on a board before the buzzer goes off and makes the board spring upward. To make things interesting, a student sets the buzzer to a random time increment anywhere between 30 and 60 seconds. Consider time until the buzzer sounds a random variable where any time between 30 and 60 has an equal likelihood.

b. Random Variable 2: An industrial process yields a large number of steel cylinders. The length of the cylinder is a random variable with an average of 5.01 centimeters and a standard deviation of 0.0046 centimeters. The distribution of cylinder lengths is symmetrical, where lengths are more likely to be close to the mean rather than further away from the mean.

c. Random Variable 3: Idaho recently experienced a 6.5 magnitude earthquake that could be felt in parts of neighboring states, including Oregon. Large earthquakes often are followed by smaller earthquakes called aftershocks. After a large earthquake, the first aftershock is more likely to occur closer in time to the initial mainshock than later. Suppose a researcher determined that the average time to the first aftershock in a particular region is 2.3 days. If X is the length of time in which the first aftershock occurs after the mainshock, which distribution would best model X?