The Maximum Likelihood Estimation (MLE) for the parameter $\lambda$ in a Poisson distribution is derived as follows:

1. Poisson Distribution Probability Mass Function (PMF)

The probability mass function (PMF) of a Poisson-distributed random variable $X$ with parameter $\lambda > 0$ is:

$P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}, \quad x = 0, 1, 2, \dots$

Here, $\lambda$ is the rate or average number of occurrences in a fixed interval.

2. Likelihood Function

Suppose we have a sample $X_1, X_2, \dots, X_n$ drawn from a Poisson distribution. The likelihood function $L(\lambda)$ is the joint probability of observing these $n$ independent values:

$L(\lambda) = \prod_{i=1}^{n} P(X_i = x_i) = \prod_{i=1}^{n} \frac{\lambda^{x_i} e^{-\lambda}}{x_i!}$

Since the values $x_1, x_2, \dots, x_n$ are independent, we multiply the individual probabilities.

$L(\lambda) = \left( \prod_{i=1}^{n} \frac{1}{x_i!} \right) \lambda^{\sum_{i=1}^{n} x_i} e^{-n\lambda}$

The first part of the expression is a constant with respect to $\lambda$, so we can focus on:

$L(\lambda) = C \cdot \lambda^{\sum_{i=1}^{n} x_i} e^{-n\lambda}, \quad C = \prod_{i=1}^{n} \frac{1}{x_i!}$

3. Log-Likelihood Function

To simplify the maximization, we take the natural logarithm of the likelihood function to obtain the log-likelihood function:

$\log L(\lambda) = \log(C) + \left( \sum_{i=1}^{n} x_i \right) \log(\lambda) - n\lambda$

Since $\log(C)$ is constant, we focus on the terms that involve $\lambda$:

$\log L(\lambda) = \left( \sum_{i=1}^{n} x_i \right) \log(\lambda) - n\lambda$

4. Differentiating the Log-Likelihood

To find the value of $\lambda$ that maximizes the log-likelihood, we differentiate $\log L(\lambda)$ with respect to $\lambda$ and set it equal to zero:

$\frac{d}{d\lambda} \log L(\lambda) = \frac{\sum_{i=1}^{n} x_i}{\lambda} - n$

Set the derivative equal to zero:

$\frac{\sum_{i=1}^{n} x_i}{\lambda} - n = 0$

Solving for $\lambda$:

$\hat{\lambda} = \frac{\sum_{i=1}^{n} x_i}{n}$

5. Conclusion: MLE for $\lambda$

The Maximum Likelihood Estimator (MLE) for $\lambda$ is the sample mean:

$\hat{\lambda} = \frac{1}{n} \sum_{i=1}^{n} X_i$

This means that the MLE for $\lambda$ is simply the average of the observed values from the Poisson-distributed sample.

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Would you like further details or have any questions on this derivation?

Here are 5 related questions you may find helpful:

1. How can we assess the variance of the MLE for $\lambda$ in a Poisson distribution?
2. What is the Fisher information for the Poisson distribution?
3. How does the MLE for $\lambda$ behave in small sample sizes for Poisson data?
4. What is the confidence interval for $\lambda$ based on the MLE in a Poisson distribution?
5. How do we compute the likelihood ratio test for comparing Poisson distributions with different $\lambda$?

Tip: MLE is often used for parameter estimation, but be mindful of sample size—larger samples provide more reliable estimates and often result in better asymptotic properties like consistency and efficiency.