Answer to Question #317210 in Statistics and Probability for sss

Question

The maximum likelihood estimate for λ

 of a Poisson distribution P(X=x)=f(x,λ)

Answer 1

$\sum_i{\ln f\left( x_i,\lambda \right)}\rightarrow \max \\l=\sum_i{\ln \frac{\lambda ^{x_i}e^{-\lambda}}{x_i!}}=\left( x_1+x_2+...+x_n \right) \ln \lambda -n\lambda -\ln x_1!x_2!...x_n!\rightarrow \max \\\frac{dl}{d\lambda}=\frac{x_1+x_2+...+x_n}{\lambda}-n\\\frac{dl}{d\lambda}=0\Rightarrow \lambda =\bar{x}\\It\,\,is\,\,\max because\,\,l\left( 0 \right) =-\infty ,l\left( +\infty \right) =-\infty \\Thus\\\hat{\lambda}=\bar{x}$