1. Suppose a random variable X is beta distributed with parameters α = β = 3.
Find X's probability density function. Then determine P(X ≥ 0.75).
2. Suppose that for a beta distribution n = 12, α = 6 and p = 0.30. Using (7.68), find the probability that X is less than or equal to p. What is the probability that X is less than or equal to 0.7?
3. Bill and Alice's Country Store has propane tanks that are filled every Wednesday. Experience indicates that the proportion of gas used between fills is beta distributed with α = 3 and β = 2. Determine the probability that at least 85% of the gas will be used between fills.
4. The proportion of defective parts produced by a metal stamping machine follows a beta distribution with α = 1 and β = 10. Find:
(a) The probability that the proportion of defective units exceeds 5%
(b) The probability that the proportion defective is more than one standard deviation above the mean.