A civil engineer is studying a left-turn lane that is long enough to hold seven cars. Let X be the number of cars in the lane at the end of a randomly chosen red light. The engineer believes that the probability that X = x is proportional to (x + 1)(8 − x) for x = 0, . . . , 7 (the possible values of X).
a. Find the p.f. of X.
b. Find the probability that X will be at least 5.

Suppose that two balanced dice are rolled, and let X denote the absolute value of the difference between the two numbers that appear. Determine and sketch the p.f. of X.

Let f0(x) be the p.f. of the Bernoulli distribution with parameter 0.3, and let f1(x) be the p.f. of the Bernoulli distribution with parameter 0.6. Suppose that a single observation X is taken from a distribution for which the p.d.f. f (x) is either f0(x) or f1(x), and the following simple hypotheses
are to be tested:
H0: f (x) = f0(x),
H1: f (x) = f1(x).
Find the test procedure δ for which the value of α(δ)+β(δ) is a minimum.