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The Poisson distribution is a powerful statistical tool used to model the number of times an event occurs within a specified time period. It's particularly useful when dealing with events that happen independently and on average a certain number of times. The formula to calculate the probability of observing exactly \( k \) events is given by:

• \( P(X = k) = \frac{e^{-\mu} \mu^k}{k!} \)
• Here, \( \mu \) is the average occurrence rate.

For example, if you are counting the number of breakdowns in a computer system, and if these breakdowns follow a Poisson distribution, \( \mu \) represents the average number of breakdowns in a given time frame. This concept is vital in the exercise, as both \( X_i \) and \( Y_i \) follow Poisson distributions with parameters \( \mu_1 \) and \( \mu_2 \) respectively. Understanding this relationship is key to solving the problem of finding the MLE for these parameters.

When we refer to independent random variables, it means that the occurrence of one event does not affect the occurrence of another. In other words, the outcome of one variable has no impact on the outcome of the other. This assumption simplifies many statistical models because it implies that no external factor links the events together.In our context, \( X_i \) and \( Y_i \) are independent random variables, indicating that breakdowns in one system are not influenced by the behavior of the other system. This property allows us to separate the likelihood functions for \( X_i \) and \( Y_i \), making it easier to compute and maximize the individual log-likelihoods, ultimately leading to the respective MLEs for \( \mu_1 \) and \( \mu_2 \). Independence is a common assumption in statistical estimation, especially when starting with fundamental models like the Poisson distribution.

The log-likelihood function is derived from the likelihood function, which measures how probable the observed data is under certain parameter values. Taking the logarithm of the likelihood function, hence the term log-likelihood, simplifies calculations, especially in the presence of products.If you multiply lots of probabilities together, things get complex quickly. By using the properties of logarithms, you convert products into sums, which are easier to handle mathematically. For instance:

• The log-likelihood for the Poisson distribution of system 1 is: \( \log L(\mu_1) = -n\mu_1 + \left( \sum_{i=1}^{n} X_i \right) \log \mu_1 - \sum_{i=1}^{n} \log(X_i!) \)

Quantifying how likely our data is given the parameter \( \mu \) allows us to derive the MLE by setting the derivative of this log-likelihood function to zero. This transformation is crucial in statistically estimating parameters efficiently.

Statistical estimation involves finding the parameter values that best describe a given set of data. One common method is Maximum Likelihood Estimation (MLE), which finds the parameters that maximize the likelihood that the observed data came from a specified statistical model.A key benefit of MLE is that it provides a method to estimate unknown parameters in a probabilistic model. In the context of the provided exercise:

• The MLE for \( \mu_1 \) is \( \hat{\mu}_1 = \frac{\sum X_i}{n} \).
• The MLE for \( \mu_2 \) comes out as \( \hat{\mu}_2 = \frac{\sum Y_i}{n} \).

When estimating the difference \( \mu_1 - \mu_2 \), the MLE simply becomes the difference between the individual estimators: \( \widehat{\mu_1 - \mu_2} = \hat{\mu}_1 - \hat{\mu}_2 = \frac{\sum X_i}{n} - \frac{\sum Y_i}{n} \). This technique makes statistical analysis both practical and insightful, allowing meaningful insights to be drawn from data.