Build and maximize a likelihood

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Problem

Observations $1,2,3$ are i.i.d. from an exponential distribution with unknown rate $\lambda>0$ . Write the likelihood and log-likelihood, then find the MLE.

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The likelihood is

L(\lambda)=\prod_{i=1}^3\lambda e^{-\lambda x_i} =\lambda^3e^{-6\lambda},\qquad \lambda>0

Thus $\ell(\lambda)=3\log\lambda-6\lambda$ . Setting $\ell'(\lambda)=3/\lambda-6=0$ gives $\hat\lambda=1/2$ . Since $\ell''(\lambda)=-3/\lambda^2<0$ , this is the unique maximum.

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Build and maximize a likelihood

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