35. The median, like the mean, is important in predicting the value of a random variable. Whereas it was shown in the text that the mean of a random variable is the best predictor from the point of view of minimizing the expected value of the square of the error, the median is the best predictor if one wants to minimize the expected value of the absolute error. That is, E[|X − c|] is minimized when c is the median of the distribution function of X . Prove this result when X is continuous with distribution function F and density function f . (Hint: Write

Now, use calculus to find the minimizing value of c.)

Transcribed Image Text:

E[|X - c]] = |x c f (x) dx -0 - 00 | |x c| f (x) dx + |x c\ f (x) dx == \ - (c-x)f(x) dx + - L 1)*] = (x-c)f (x) dx = c(c) - | f (x) dx + x + xf (x) dx c[1 F(c)] -00 -