Thread: Two questions

(1) show that -log(U) is exponential(1) if U is uniform(0,1)

(2)Let X be exponential(2). Set Y=1-exp(-2Y) . Show Y is uniform(0,1)

I thinks these two are quite similar. But I still dont know how to solve it. Anyone can? thx.

Originally Posted by solskjaer

(1) show that -log(U) is exponential(1) if U is uniform(0,1)

[snip]

Let $\displaystyle Y = - \ln U$ and find the cdf of Y:

$\displaystyle F(y) = \Pr(Y \leq y)$

noting that $\displaystyle 0 \leq u \leq 1 \Rightarrow 0 \leq y \leq +\infty$

$\displaystyle = \Pr(- \ln U \leq y) = \Pr(\ln U \geq - y) = \Pr(U \geq e^{-y}) = \int_{e^{-y}}^1 1 \, du$ $\displaystyle = 1 - e^{-y}$.

Then $\displaystyle f(y) = \frac{dF}{dy} = e^{-y}$ for $\displaystyle 0 \leq y \leq +\infty$ and zero elsewhere, as required.


You should re-try (2).