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

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.

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1 for 0x1 f(x) = 10 for all other x