Final answer:
The random variable X would represent the number of IRS audits per year in a sample of 100 people who all have tax returns over $25,000. It is modeled by a Poisson distribution with a mean of 2 for one year. Over 20 years, the distribution of X is the sum of 20 independent Poisson-distributed variables.
Step-by-step explanation:
The question pertains to the probability of being audited by the Internal Revenue Service (IRS) and the application of a Poisson distribution to model this scenario. The random variable X in this context would represent the number of people audited by the IRS in one year out of a randomly picked sample of 100 people who all have tax returns reporting more than $25,000 in income. Since the rate of audit is given as approximately 2 percent per year, the values that X may take on would theoretically be any non-negative integer starting from 0, with the distribution being Poisson with a mean λ equal to 2 (2 percent of 100 people).
For the 20-year period question, the random variable X would define the total number of audits a person with income over $25,000 is expected to experience over 20 years, assuming each year is independent of the others. The values that X can take on range from 0 up to 20 (or possibly higher, depending on the specifics of multiple audits within a single year), with each year's audit probability being independent. Therefore, the distribution of X over a 20-year period can be calculated using the sum of independent Poisson-distributed variables, which is again a Poisson distribution with parameter λ multiplied by 20 (the number of years).