Help Finding Median in Probability?

Question

pdf(x)=1/(x ln(1.5)) for 4<=x<=6, pdf(x)=0 elsewhere.

Trying to find the median, which is supposed to be F(x)=.5. F(x) (as far as I know) is the integral of f(x), so in this case, is ln(x)/ln(1.5). When I solve ln(x)/ln(1.5)=.5 though, I get 1.2247, which is outside the range of possible values (4 to 6).

What have I done wrong? And did whatever I did wrong affect my attempt to find expected value (I got 4.93, which I think is correct)?

Answer 1

In order to find the median, you have to set the _definite_ integral form -∞ to x = 0. You set the indefinite integral equal to zero, and implicitly assumed in the process that the pdf is 1/(x ln 1.5) for all x, as opposed to 4≤x≤6. Now the definite integral of a piecewise function is given by the sum of the integrals of each of the pieces in the range of the integral - that is (-∞, x)∫pdf(x) dx = (-∞, 4)∫0 dx + (4, x)∫1/(x ln 1.5) dx. The first integral is obviously zero, and the second is given by ln x/ln 1.5 - ln 4/ln 1.5. Note the difference between that and the indefinite integral. Setting this equal to 1/2 and solving:

(ln x)/(ln 1.5) - (ln 4)/(ln 1.5) = 1/2
ln x = 1/2 ln 1.5 + ln 4
x ≈ 4.898979

Which is a more reasonable answer.