Word Count : 358

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Let (X_1, X_2, \dots, X_n) be a random sample from a Poisson distribution with parameter (\lambda), representing the number of breakdowns per month. The probability mass function is given by (P(X_i = x_i) = \frac{e^{-\lambda} \lambda^{x_i}}{x_i!}), where (x_i = 0,1,2,\dots). The likelihood function for the sample is [ L(\lambda) = \prod_{i=1}^n \frac{e^{-\lambda} \lambda^{x_i}}{x_i!} = e^{-n\lambda} \lambda^{\sum x_i} \prod_{i=1}^n \frac{1}{x_i!}. ] Taking logarithms, the log-likelihood function is [ \ell(\lambda) = -n\lambda + _____ __________ __________ _____ ____ ___ __________.
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