7 The median of a continuous random variable Y with cdf FY y

7. The median of a continuous random variable Y with cdf FY (y) is the value m such that FY (m) = P(Y m) = 0.5. That is, Y is just as likely to be larger than its median as it is to be smaller. If fY (y) denotes the pdf of Y , then we know that m solves the following equation 0.5 = FY (m) = Z m fY (y)dy; that is, the area under fY (y) to the left of m is 0.5 and the area to the right of m is also 0.5. For each of the distributions, calculate the median m.

(a) Y Unif(0, ), where > 0.

(b) Y Exp(), where > 0.

c) Y N(µ, 2 ), where > 0.

(d) Y Beta(, ), where = > 0.

Solution