Question

4. The Median Now that you have had some experience with cost functions, try minimizing the expectation value of this cost function: x-i. This is not as nice, since there is an absolute value, which is discontinuous. Show that the result of minimizing the expected value of fx- with respect to your estimate is the median. Hint: Break the integral into two pieces using the definition of the absolute value Hint 2: You don't have to show that one-half of the integral is 50%. You only need to show that the probability to the left of the median is equal to the probability to the right of the median.

Answer 1

Solution: Lex yol yTagline.de Since Tonal. $(x- *'); -for en å |-(x -^); for xc â Hence from cquation 0, y can be witten ag - 7. fln) dn x fcx du 4 11x. ax) fox) dx + 11 ley = h) f(x) dx crorential differentiating it coith respect to â , we have o b foo) da J (2)

feld f(a)dn en este post froste. Jos also, o ( ) - â = median of x al ) Since f(x) do - I falda feos do- -Boost few da I faldx . 2. f(x) dx - 1 de = 2. FCT) - 1 Thus metus (2.F(3)-1) = 24() > 0 at median of x nos y = E12- ôl is maximum and Since day so dâ)2 dyo gives a = median of it