Question

Annual incomes for heads of household in a section of a city have approximately a gamma distribution with α=20 and β=1000. a Find the mean and variance of these incomes. b Would you expect to find many incomes in excess of $30,000 in this section of the city?

Answer 1

The mean and variance of annual incomes for heads of household in a section of a city can be found using the parameters of the gamma distribution. To determine if there are many incomes in excess of a certain amount, the probability can be calculated using the cumulative distribution function of the gamma distribution.

Step-by-step explanation:

a) The mean of a gamma distribution with parameters α and β is given by α * β and the variance is given by α * β^2.

In this case, α = 20 and β = 1000, so the mean is 20 * 1000 = 20,000 and the variance is 20 * 1000^2 = 20,000,000.

b) In order to determine whether there are many incomes in excess of $30,000, we need to calculate the probability that an income is greater than $30,000.

This can be done by finding the cumulative distribution function (CDF) of the gamma distribution and evaluating it at $30,000.

If the resulting probability is high, then we would expect to find many incomes in excess of $30,000.