Question

Suppose X follows a Gamma distribution with parameters α, β, and the following density function f(x)= [x^(α−1)e^(−x/ β)]/ Γ(α)β^α . Find α and β so that E(X)= Var(X)=1. Also find the median for the random variable, X.

Answer 1

Solution:-

Let X~Gamma (α, β) distribution

Therefore, pdf of X is given as,

Mean and variance of this distribution is,

Therefore, consider ratio,

But we have given that,

So, from above ratio,

Now use equation (i) in E(X)

Therefore,

#Find median of the distribution:-

-----> Now, first put value of (α=1,β=1) in given pdf

Therefore, the pdf of X is written as,

That is,

Now, by definition of Median, Median is given by

Therefore,

Taking natural logarithm { } on both sides,we get

Result:-