In practical applications, we may be able to assume that the conditions for a Poisson process hold at least approximately, but the rate of events, \(\lambda\), is usually an unknown value that must be estimated from data.

Given a random sample, \(\{x_1, x_2, \dots, x_n\}\), from a \(\PoissonDistn(\lambda)\) distribution, we will use maximum likelihood to estimate \(\lambda\). The logarithm of the Poisson probability function is

\[ \log(p(x | \lambda)) \;=\; x \log(\lambda) - \lambda - \log(x!) \]

so the log-likelihood function is

\[ \ell( \lambda) \;=\; \sum_{i=1}^n {x_i} \log(\lambda) - n\lambda + K \]

where \(K\) is a constant that does not depend on \(\lambda\). To find the maximum likelihood estimate, we solve

\[ \ell'( \lambda) \;=\; \frac {\sum {x_i}} {\lambda} - n \;=\; 0 \]

so

\[ \hat{\lambda} \;=\; \frac {\sum {x_i}} n \;=\; \overline{x} \]