We are studying the earthquake energy that the villages near the earthquake center can receive. There are many villages located along the line with minimum -1 and maximum +2 kilometers away from the earthquake center (see the picture below for illustration). Suppose we randomly pick one village in that area. That is, if we let the location of the earthquake center be 0, the location of the randomly picked village is uniformly distributed in the interval [-1, 2]. Also, suppose that the earthquake energy that the picked village can receive is 1 plus 25% of its squared location.

Figure 1: Schematic diagram

(a) What is the cumulative distribution function of the earthquake energy that a randomly picked village can receive?

(b) What is the probability density function of the earthquake energy that a randomly picked village can receive?

(c) What is the expected amount of the earthquake energy that a randomly picked village can receive?

-1 -0.5 Village: Location=-0.5 Earthquake center 1.5 2 Village: Location=1.5