Determining the distribution of univariate transformation

Question

If Y is uniformly distributed on the interval $(0, 1)$ and if

$Z = –a * ln(1 – Y)$ for some $a > 0$,

then to which of the following families of distributions does Z belong?

1. Lognormal
2. Exponential
3. Uniform
4. Cauchy
5. Normal

Attempt:

$\frac{Z}{-a} = ln(1-Y)$

$\large e^{Z/-a} = 1-Y$

$\large 1-e^{z/-a} = Y$

$\large \frac{\delta}{\delta z}[1-e^{z/-a}] = \frac{-1}{a}$

$\large (1-e^{z-a})*|\frac{1}{a}| = \frac{1}{a}-\frac{1}{a}e^{(-z/a)}$

Exponential

Answer 1

We go for the cdf of $Z$. For positive $z$ we have $$\Pr(Z\le z)=\Pr(-\ln(1-Y)\le z/a)=\Pr(\ln(1-Y)\ge -z/a)=\Pr(1-Y\ge e^{-z/a}).$$ This is $\Pr(Y\le 1-e^{-z/a})$, which is $1-e^{-z/a}$. Conclusion: exponential.