5.5 An analyst has available two forecasts, F1 and F2, of earnings per share of a corporation next year. He intends to form a compromise forecast as a weighted average of the two individual forecasts. In forming the compromise forecast, weight X will be given to the first forecast and weight 11 - X2, to the second, so that the compromise forecast is XF1 + 11 - X2F2.

The analyst wants to choose a value between 0 and 1 for the weight X, but he is quite uncertain of what will be the best choice. Suppose that what eventually emerges as the best possible choice of the weight X can be viewed as a random variable uniformly distributed between 0 and 1, having the probability density function f1x2 = e 1 for 0 … x … 1 0 for all other x

a. Graph the probability density function.

b. Find and graph the cumulative distribution function.

c. Find the probability that the best choice of the weight X is less than 0.25.

d. Find the probability that the best choice of the weight X is more than 0.75.

e. Find the probability that the best choice of the weight X is between 0.2 and 0.8.