Question

 Assume that X has the exponential distribution with parameter : Is it possible to find a function G .x/ such that Y D G .X/ has uniform distribution over [0; 1]? Justify your answer!

Answer 1

The exponential distribution has probability density function $\lambda e^{-\lambda Y}$

 and cumulative distribution function (CDF) $1-e^{-\lambda Y}$

We map the unit interval onto the range of the CDF. Then, inverting the CDF, we get

$Y=-\frac{ln(1-X)}{\lambda}$ for 0≤X<1 (we can ignore 1 because it occurs with probability 0).

So the function is (with λ=1)

$g(X)=−ln(1−X)$