# The pdf of sum of -log($U_i$) in which Ui is iid uniform distributed

Suppose $$Ui$$ is independently uniformed distributed between [0,b], $$Y = -\Sigma_1^n log(U_i)$$. what is the pdf of Y? I tried used characteristic function but it doesn't match each of usual distribution.

Note: the following argument assumes $$b=1$$. To generalise, add $$\ln b$$ to each $$\ln U_i$$ term, i.e. $$-n\ln b$$ to $$y$$ so its pdf shifts.
You probably already worked out $$-\ln U_i\sim\operatorname{Exp}(1)$$, because $$P(-\ln U\le x)=P(U\ge\exp -x)=1-\exp -x.$$Of course, this implies $$-\ln U_i$$ has characteristic function $$1/(1-it)$$, so $$Y$$ has cf $$1/(1-it)^n$$. Now, what distribution is that? Spoiler: it's
a Gamma distribution with $$k=n,\,\theta=1$$, so the pdf is $$\frac{y^{n-1}}{(n-1)!}\exp -y$$ for $$y\ge 0$$.
• Thanks. I have already worked out the case when $b=1$, however I couldn't figure out the generalised part. Why can we just add $-nlnb$ to y? How can you prove it is right regularization? – T.y Jan 19 at 9:11
• @T.y Because $U_i\sim U(0,\,b)$ iff $U_i/b\sim U(0,\,1)$. – J.G. Jan 19 at 9:13
• @T.y Scaling $U_i$ to $bU_i$ changes the characteristic function $\varphi(t)$ to $\varphi(bt)$, not $b^{it}\varphi(t)$. – J.G. Jan 19 at 9:32