Let Y be exponentially distributed with mean 3 a Use the met

Let Y be exponentially distributed with mean 3. (a) Use the method of distribution functions to find the density function of Y². (b) Find a transformation G such that G(Y ) is uniformly distributed on (1, 2).

Solution

Density function of Y = exponential distribution with parameter 1/3

= 1/3 e^-y/3, y>=0

If we have to find a transformation such that G(y) is uniformly distributed in [1,2] means

f(G) = 1, in [1,2]

Hence G(y) = integral of pdf of y dy