structural:summarization
Example 2

Suppose X {\displaystyle X} is a random variable with a cumulative distribution

where θ > 0 {\displaystyle \theta >0} is a fixed parameter. Consider the random variable Y = l o g (1 + e − X). {\displaystyle Y=\mathrm {log} (1+e^{-X}).} Then,

The last expression can be calculated in terms of the cumulative distribution of X, {\displaystyle X,} so

which is the cumulative distribution function (CDF) of an exponential distribution.