Density of a scaled log transformed random variable?

Question

If $X$ and $Y$ are two independent random numbers from the interval $(0,1)$, then what is the PDF of $U = -2 \ln X$ and $V = -2\ln Y$?

Answer 1

A general way for solving this type of problem is to find the CDF of the transformed variable in terms of the CDF of the original variable, then take the derivative to find the PDF of the transformed variable.

For example, assume $X$ is a random variable taking values in $(0,1)$ with PDF $f_X(x)$ and CDF $F_X(x)$. We would like to know the CDF of $U=-2\log(X)$.

$$P(U\leq u)=P(-2\log(X)\leq u)=P(X\geq e^{-\frac{u}{2}})=1-F_X(e^{-\frac{u}{2}})$$

Now we differentiate the CDF of $U$ with respect to $u$ and find that the PDF of $U$ is the following.

$$f_U(u)=\frac{d}{du}\left(1-F_X(e^{-\frac{u}{2}})\right)=-f_X\left(e^{-\frac{u}{2}}\right)\left(-\frac{1}{2}e^{-\frac{u}{2}}\right)=\frac{1}{2}e^{-\frac{u}{2}}f_X\left(e^{-\frac{u}{2}}\right)$$

Because $X$ takes on values between $0$ and $1$, $U$ takes on values between $0$ and $\infty$.

The same process can be completed for $V=-2\log(Y)$.

You can identify the distributions of $U$ and $V$, if they are well known, by inspection of the PDF or CDF.

Answer 2

If $X$ and $Y$ are independent uniform $U$ and $V$ are independent negative exponential with rate parameter $1/2$. You can solve this by change of variables.