Thread: Likelihood function of gamma distribution

Let X₁, X₂,...X_n be i.i.d. Gamma(a,b) random variables where both a and b are unknown. Let T₁= (X₁ + X₂ +...+ X_n)/n and T₂ = (X²₁ + X²₂ +...+ X²_n)/n be the first and second sample moments, respectively. Show the likelihood function of a and b.

So...I don't really understand what a likelihood function is. I believe it is the same algebraically as f(x₁|a,b)*f(x₂ | a,b)*...*f(x_n | a,b), but I don't really know how to find this in terms of a and b. Any help is much appreciated. Thanks so much!

Hey Aria1.

A likelihood function is just a probability function that is often given conditional on various sample values.

A typical likelihood function is a product of likelihood functions that correspond to a particular sample.

What you need to do first is setup the likelihood function and then find the log-likelihood function.

So to start off, can you first name the probability function for P(X = x) for a Gamma distribution?

What I was thinking was that the likelihood function is the (pdf of gamma)^n. So, the probability function is: f(x1,x2,...xn|a,b) = [(b^a)/gamma(a)]*x^(a-1)e^(-bx). Then, I would get the likelihood function to be L(a,b | x1, x2,...,xn)= f(x1,x2,...xn|a,b) ^n where x1...xn are represented in summation form. Is this the right direction to take it? I also don't really know what the support of that function might be. I think a and b are > 0?

You have the right idea, but remember that all realizations (i.e. the x's) are different values that correspond to the sample that you are using to estimate your parameter.