Question

a. What is the maximum likelihood estimator for the parameter 2 of the poisson distribution for a sample of n poisson random variables?

Answer 1

l ( $\lambda$ ; x) = $\prod$ e^{- $\lambda$} $\lambda$ ^X / X1

= e^{-n $\lambda$} $\lambda$ ^{$\sum$ Xi} / Xi !

Taking log on both sides

ln (l( $\lambda$ ; x) ) = - n $\lambda$ + $\sum$ Xi ln $\lambda$ + ln (xi ! )

Take deriavative and equate to 0

d/dx ( ln (l( $\lambda$ ; x) )) = - n + $\sum$ Xi / $\lambda$ + 0

- n + $\sum$ Xi / $\lambda$ = 0

$\sum$ Xi / $\lambda$ = n

Solve for $\lambda$

$\lambda$ = $\sum$ Xi / n

Thereofore,

Maximum likelihood estimator of $\lambda$ is

$\hat{\lambda }$ = $\sum$ Xi / n ( OR $\hat{\lambda }$ =   $\bar{x}$ )