Show that a continuous probability distribution that is memoryless must be exponential. Hint: For g(t)=P(X>t), show that g(t)= (g (1)) t for all positive, rational t. Starting at noon, diners arrive...

Show that a continuous probability distribution that is memoryless must be exponential. Hint: For g(t)=P(X>t), show that g(t)= (g (1))
^t
for all positive, rational t.

Starting at noon, diners arrive at a restaurant according to a Poisson process at the rate of five customers per minute. The time each customer spends eating at the restaurant has an exponential distribution with mean 40 minutes, independent of other customers and independent of arrival times. Find the distribution, as well as the mean and variance, of the number of diners in the restaurant at 2 p.m.