Math Help - Gamma variables and transformations

1. Gamma variables and transformations

If Y is uniformly distributed over the interval (0,1), i.e. f(y) = 1 for 0< y < 1.
Show that U = -2 loge(Y) has a negative exponential distribution. By comparing the density
functions or the moment generating functions, show also that an exponential distribution is
identical to a chi-square distribution with 2 degrees of freedom. If Y1, Y2, ..., Yn are
independently uniformly distributed over the interval (0,1), determine the distribution of
U = -2loge(Y1Y2...Yn) = summation(-2loge(Yi)).

2. Originally Posted by sonitgarg
If Y is uniformly distributed over the interval (0,1), i.e. f(y) = 1 for 0< y < 1.
Show that U = -2 loge(Y) has a negative exponential distribution. By comparing the density
functions or the moment generating functions, show also that an exponential distribution is
identical to a chi-square distribution with 2 degrees of freedom. If Y1, Y2, ..., Yn are
independently uniformly distributed over the interval (0,1), determine the distribution of
U = -2loge(Y1Y2...Yn) = summation(-2loge(Yi)).
Here's a start:

Calculate the cdf of U:

$F(u) = \Pr(U < u) = \Pr(-2 \ln Y < u) = \Pr\left(\ln Y > -\frac{u}{2}\right)$ $= \Pr(Y > e^{-u/2}) = \int^{1}_{e^{-u/2}} dy = ....$ if u > 0 and zero otherwise.

Now recall that the pdf of U is given by $f(u) = \frac{dF}{du}$.