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  • STA260 Lecture 08

MLE for Poisson Parameter

• Setup:

  • $X_1,\dots ,X_n\sim \mathrm{Poisson}(\lambda )$.
  • PDF: $f(x;\lambda )=\frac{e^{-\lambda }{\lambda }^x}{x!}$.

• Likelihood Function:

  • $L(\lambda )=\prod _{i=1}^n\frac{e^{-\lambda }{\lambda }^{x_i}}{x_i!}=\frac{e^{-n\lambda }{\lambda }^{\sum x_i}}{\prod x_i!}$.

• Log-Likelihood:

  • $\ell (\lambda )=-n\lambda +(\sum x_i)\ln (\lambda )-\ln (\prod x_i!)$.

• Maximization:

  • Differentiate w.r.t. $\lambda$:
    • $\frac{d}{d\lambda }\ell (\lambda )=-n+\frac{\sum x_i}{\lambda }$.
  • Set to zero:
    • $-n+\frac{\sum x_i}{\hat{\lambda }}=0$.
    • $n=\frac{\sum x_i}{\hat{\lambda }}$.

• Result:

  • ${\hat{\lambda }}_{[[pages/MaximumLikelihoodEstimation(MLE)Principle\parallel MLE]]}=\frac{\sum x_i}{n}=\overline{X}$.

• Footnotes