If $U$ has uniform distribution over the interval $(0, 1)$ what is the density function of $X = -Kln(U)$ for some constant $K > 0$?

Question

I am aware of he uniform distribution density function however I am not entirely sure how I would find out the distribution of X.

Thanks very much, any help or hints in the right direction would be gladly appreciated.

You can use the CDF method to solve this. See http://www.maths.qmul.ac.uk/~pettit/MTH5122/chapter2.pdf — Jürgen Sukumaran, Aug 03 '16 at 19:01

score 2 · Accepted Answer · answered Aug 03 '16 at 19:09

We start by looking at the cumulative distribution function for $X$, \begin{equation} F_X(x) =P(X\leq x) =P(-K\ln U\leq x) = P(\ln U \geq -\frac{x}{K})=P(U\geq e^{-\frac{x}{K}}) \end{equation} Now we will use the pdf of the uniform distribution and integrate. This will give you the CDF of $X$ explicitly which you can differentiate to find the pdf of $X$.

score 1 · Answer 2 · answered Aug 03 '16 at 19:08

1

HINT: If we define $F(x) = \mathbb{P}(X \leq x)$, then the density of $X$ is the derivative of $F(x)$. Can you find a nice form for $F(x)$ using the definition of $X$?

answered Aug 03 '16 at 19:08

Marcus M

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If $U$ has uniform distribution over the interval $(0, 1)$ what is the density function of $X = -Kln(U)$ for some constant $K > 0$?

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