# Maximum likelihood estimators

#### NixLUKE

Hello, im struggling with setting up the likelihood function from the probability function given. I just do not understand the rules governing how things are mulitplied etc. for example

#### chiro

MHF Helper
Hey NixLUKE.

Hint - If all sample elements are independent and identically distributed you use P(A=a and B=b) = P(A=a) x P(B=b).

If you do that for all sample elements and use the IID property (they all have the same distribution with the same parameters) then you can get the likelihood function of the sample.

1 person

#### romsek

MHF Helper
the resulting pdf of a vector of $N$ observations, $\pmb{x}$ will be the product of the individual pdfs, i.e.

$f_{\theta}(\pmb{x},p)=\displaystyle{\prod_{n=1}^N} \dfrac{x_n^{p-1}e^{-\frac{x_n}{\theta}}}{\theta^{Np} ~~ \left( \Gamma(p) \right)^n}$

to find the maximum likelihood estimate of $\theta$ take the derivative with respect to $\theta$ of $\ln\left(f_{\theta}(\pmb{x},p) \right)$

set it equal to $0$ and solve for $\theta$

The resulting MLE is a pretty simple form.

1 person