Solved – Rayleigh Distribution Quartiles

The Rayleigh distribution has PDF f(x) =xe−$\frac{x^2}{2}$, x >0. Let X have the Rayleigh distribution.

(a) Find P(1< X < 3).

(b) Find the first quartile, median, and third quartile of X.

Alright, so the first part is quite easy– it's just the integral from 1 to 3 of f(x), but the second part is tricky. I know F(x) = 1 – e$^\frac{-x^2}{2}$, but the inverse is a function that doesn't exist. Any help?

The inverse of $F$ exists: You have to use Naperian logarithm, i.e., $ln (e^a) = log_e (e^a) = a$.