Maximum likelihood estimator

Question

Let x1, . . . , xn be a random sample of a Poisson distributed population X ∼ Po(λ) with λ > 0. Find the maximum likelihood estimator for λ.

Answer 1

If we have a sample x=(x₁,x₂,...,x_n) then the likelihood is L(λ|x) = Π_i (e^-λλ^xi/x_i!).

The log-likelihood is l(λ|x) = Σ_i ln(e^-λλ^xi/x_i!) = Σ_i [ln(e^-λ) + ln(λ^xi) - ln(x_i!)] = -nλ + (Σ_i x_i)ln(λ) - Σ_i ln(x_i!)

Differentiating gives: dl/dλ = -n + (Σ_i x_i)/λ = -n + n*mean(x)/λ.

Setting this to zero and solving gives λ_MLE = mean(x).

So the MLE for λ is just the sample mean of the data.