5.15 The weekly repaik cost, X, for a certain machine has a probability density function given by 6x(1 - x), 0

5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?

The probability density function of a distribution is given by x f(x) = exp(-7). Use differentiation to show that the probability density function has a maximum at x = 0. The moment generating function of a distribution is M(t) = (q + pet)", where p € [0, 1], q = 1 - p and n is a positive integer. Use the moment generating function to find the mean and variance of the distribution in terms of p, q and n.