If X Uniform0 1 show that Y 5 In X has a Gamma distributio

If X ~ Uniform(0, 1), show that Y = -5 In X has a Gamma distribution, identify the parameters.

Solution

Given that X is a uniform random variable in (0,1)

Then we know that z = -ln X is a gamma variable with parameters 1 and 1.

i.e. f(z,1,1) = e^-z, z>=0

Y = -5ln x = 5z

Hence distribution of Y will be

f(y) = (1/5) e^-y/5, y>=0

Hence parameters are alpha = 4/5 and beta = 1