1. There are 26 students in a Pre-K class.

Each of these students is either four or five years old, and 25 of these students have the letter A somewhere in their first name. Is this a statistical oddity?

2. An actuary models the lifetime of a device using the random variable Y = 10X0.8, where X is an exponential random variable with mean 1 year. Determine the probability density function f(y), for y > 0, of the random variable Y.

3. The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 250. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (0, 1500). Determine the standard deviation of the insurance payment in the event that the automobile is damaged.

4. An insurance company insures a large number of drivers. Let X be the random variable representing the company losses under collision insurance, and let Y represent the company looses under liability insurance. X and Y have joint density function

What is the probability that the total loss is at least 1?

5. A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is 3/5. The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months. Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.

6. Claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000. What is the probability that the average of 25 randomly selected claims exceeds 20,000?