A random variable X is uniformly distributed in the interval

A random variable X is uniformly distributed in the interval [0,1]. Find the pdf of the random variable Y = - lnx.

Solution

First lets find the cumulative distribution function, cdf.

Pr(Y <= y) becomes

Pr(-ln(x) <= y)

Pr(ln(x) >= -y)

Pr(x >= e^(-y))

The pdf you seek, is the differential of uniform c.d.f., e^(-y)

(1 / (1 - 0)) * d/dy(e^(-y))

-e^(-y)

Replace y with x :

PDF = -e^(-x) ---> ANSWER