# Stochastic Process, Random Variables, Discrete Random Variables, and Probability Distributions

Please do only 1, 2, 6, 7, 9, 10, 11.

1. Define (a) stochastic process; (b) random variable; (c) discrete random variable; and (d) probability distribution.

2. Without using formulas, explain the meaning of (a) expected value of a random variable; (b) actuarial fairness; and (c) variance of a random variable.

6. (a) What are the parameters of a binomial distribution? (b) What is the mean of a binomial distribution? The standard deviation? (c) When is a binomial skewed right? Skewed left? Symmetric? (d) Suggest a data-generating situation that might be binomial.

7. (a) What are the parameters of a Poisson distribution? (b) What is the mean of a Poisson distribution? The standard deviation? (c) Is a Poisson ever symmetric? (d) Suggest a data-generating situation that might be Poisson.

9. (a) When are we justified in using the Poisson approximation to the binomial? (b) Why would we want to do this approximation?

10. (a) Explain a situation when we would need the hypergeometric distribution. (b) What are the three parameters of the hypergeometric distribution? (c) How does it differ from a binomial distribution?

11. When are we justified in using (a) the Poisson approximation to the binomial? (b) The binomial approximation to the hypergeometric?

#### Solution Preview

1. Define (a) stochastic process; (b) random variable; (c) discrete random variable; and (d) probability distribution.

(a) A stochastic process is the counterpart to a deterministic process in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time, in a stochastic or random process there is some indeterminacy in its future evolution described by probability distributions.

It may also be defined as a statistical process involving a number of random variables depending on a variable parameter (usually time).

(b) A random variable is a function, which assigns unique numerical values to all possible outcomes of a random experiment under fixed conditions.

(c) A random variable is which can assume only a countable number of distinct values such as 0, 1, 2, 3, ... is called a discrete random variable.

(d) Probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It is also called probability mass function or simply probability function.

2. Without using formulas, explain the meaning of (a) expected value of a random variable; (b) actuarial ...

#### Solution Summary

The solutions to the questions are explained in a step-by-step process and the methods are explained.