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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 fairness; and (c) variance of a random variable. (a) The expected value (also known as population mean) of a random variable is its average or central value. and is useful in the analysis of the variable's distribution. (b) This is a term used in Economics or Business and involves the basic idea that you pay according to your risks. (c) The variance (or population variance) of a random variable gives an idea of how widely spread the values of the random variable are likely to be. On an average, the greater the variance, the more scattered the data. 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 datagenerating situation that might be binomial. (a) The parameters are: n (> 0) and p (0 p 1) (b) Mean, = n * p and Standard deviation, = [n * p * (1 – p)] (c) The binomial distribution is skewed to the right (positive skew) when p < 0.5, skewed to the left (negative skew) when p > 0.5, and symmetric when p = 0.5. (d) Rolling a die numbered 1 – 6 and noting down the number that faces up. 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. (a) (0 < < ∞) is the only parameter of the Poisson distribution. (b) Mean, = and Standard deviation, = (c) The Poisson distribution is generally skewed but tends to symmetry when the mean becomes large. (d) Counting the number of misprints in a book. 9. (a) When are we justified in using the Poisson approximation to the binomial? (b) Why would we want to do this approximation? (a) When n is very large but the value of p is very small, the Poisson approximation to the binomial distribution is used. (b) Consider a situation in which in a sample of n = 2000 subjects, k =8 are observed to display the characteristic, whereas only np = 3 are expected to display it. Here n is very large, but np = 3 p is very small. In this case, we may need to use the Poisson approximation to the binomial. 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? (a) We need to use the hypergeometric distribution when we have to determine the number of successes in a sequence of n draws from a finite population without replacement. Here is an example: “ABC Computers manufactures personal computers at 2 plants. The Texas plant has 40 employees, the Hawaii plant has 20. A random sample of 10 employees is to be asked to fill out benefit questionnaires. What is the probability that none of the employees in the sample work at the Hawaii plant? (b) The parameters are: N (0, 1, 2), n (0, 1, 2, … N) and m (0, 1, 2, … N) (c) In a hypergeometric distribution since the draws are without replacement, the probability of success varies with trials. In a binomial distribution, the probability of success is constant in every trial. 11. When are we justified in using (a) the Poisson approximation to the binomial? (b) The binomial approximation to the hypergeometric? (a) When n is very large but the value of p is very small, the Poisson approximation to the binomial distribution is used. (b) When N >> n, we use the binomial approximation to the hypergeometric.