Problem 4 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

In Section 19.1, we considered the arrivals of packages at a network server, where we modeled the number of arrivals per minute by a Pois \((\mu)\) distribution. Let \(x_{1}, x_{2}, \ldots, x_{n}\) be a realization of a random sample from a Pois \((\mu)\) distribution. We saw on page 286 that a natural estimate of the probability of zeros in the dataset is given by \(\frac{\text { number of } x_{i} \text { equal to zero }}{n}\). a. Show that the likelihood \(L(\mu)\) is given by $$ L(\mu)=\frac{\mathrm{e}^{-n \mu}}{x_{1} ! \cdots x_{n} !} \mu^{x_{1}+x_{2}+\cdots+x_{n}} . $$ b. Determine the loglikelihood \(\ell(\mu)\) and the formula of the maximum likelihood estimate for \(\mu\). c. What is the maximum likelihood estimate for the probability \(e^{-\mu}\) of zero arrivals?

Step-by-Step Solution

Answer

The MLE for \(\mu\) is \(\frac{x_1 + x_2 + \cdots + x_n}{n}\) and for zero arrivals is \(e^{-\frac{x_1 + x_2 + \cdots + x_n}{n}}\).

Step 1: Understanding Poisson Distribution

In Poisson distribution, the probability of observing \(k\) events in a fixed interval is given by \(P(X=k) = \frac{\mu^k e^{-\mu}}{k!}\), where \(\mu\) is the mean or expected value of the distribution. Here, \(x_1, x_2, \ldots, x_n\) are realizations from a Poisson distribution with mean \(\mu\).

Step 2: Deriving the Likelihood Function

The likelihood function \(L(\mu)\) for the Poisson distribution is the product of the probabilities:\[ L(\mu) = \prod_{i=1}^{n} \frac{\mu^{x_i} e^{-\mu}}{x_i!} = e^{-n\mu} \cdot \frac{\mu^{x_1 + x_2 + \cdots + x_n}}{x_1! \cdots x_n!} \] This expression represents the likelihood \(L(\mu)\) given in the problem.

Step 3: Calculating the Log-Likelihood Function

To find the log-likelihood \(\ell(\mu)\), take the natural logarithm of the likelihood \(L(\mu)\):\[ \ell(\mu) = \ln\left(L(\mu)\right) = \ln\left(\frac{e^{-n\mu}\mu^{x_1+x_2+\cdots+x_n}}{x_1! \cdots x_n!}\right) \]This simplifies to:\[ \ell(\mu) = -(n\mu) + (x_1 + x_2 + \cdots + x_n)\ln(\mu) - \ln(x_1! \cdots x_n!) \]

Step 4: Finding Maximum Likelihood Estimate (MLE) for \(\mu\)

To find the MLE, compute the derivative of \(\ell(\mu)\) with respect to \(\mu\), then set it to zero:\[ \frac{d\ell(\mu)}{d\mu} = -n + \frac{x_1 + x_2 + \cdots + x_n}{\mu} = 0 \]Solving for \(\mu\) gives:\[ \hat{\mu} = \frac{x_1 + x_2 + \cdots + x_n}{n} \]This is the formula for the maximum likelihood estimate of \(\mu\).

Step 5: Evaluating MLE for Probability of Zero Arrivals

The probability of zero arrivals is given by \(e^{-\mu}\). Substituting the MLE \(\hat{\mu}\) for \(\mu\), the estimated probability is:\[ e^{-\hat{\mu}} = e^{-\frac{x_1 + x_2 + \cdots + x_n}{n}} \]This represents the MLE for the probability of observing zero arrivals in the dataset.

Key Concepts

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a fundamental concept in statistics. It helps us find the parameters that make the observed data most probable. In simple terms, MLE is a method for estimating the parameters of a statistical model. For example, in a Poisson distribution scenario, the parameter we aim to estimate is the mean, denoted as \( \mu \).
MLE involves finding the parameter values that maximize the likelihood function, which measures how likely it is to observe the given data under different parameter values. Once the maximum point is reached, the parameters at this point are considered the MLE.
In our exercise, the maximum likelihood estimate (MLE) for \( \mu \) is the average of our sample data, which is a very intuitive estimate when you think about it: \[ \hat{\mu} = \frac{x_1 + x_2 + \cdots + x_n}{n} \] By maximizing the likelihood, we find an estimate of \( \mu \) that best explains the number of arrivals at the network server.

Likelihood Function

The likelihood function is a critical part of the MLE process. It takes observed data and a parameter to return a value, known as the likelihood, which indicates the "fit" of the parameter to the data.
In a Poisson distribution, the likelihood function \( L(\mu) \) for a dataset from this distribution is given by:

• The probability of observing the data is modeled as a product of probabilities for each observation. Each observation follows the Poisson probability: \( P(X = x_i) = \frac{\mu^{x_i} e^{-\mu}}{x_i!} \).
  
• The likelihood is the product of these individual probabilities: \[ L(\mu) = e^{-n \mu} \cdot \frac{ \mu^{x_1 + x_2 + \cdots + x_n} }{x_1! \cdots x_n!} \]

By finding the maximum of this function, we get insights into the value of the parameter that "best" fits our data. It connects the theory of our model to the reality of observed data.

Log-Likelihood Function

Sometimes it can be simpler to work with logs rather than raw probabilities. That's where the log-likelihood function comes in. It's simply the natural logarithm of the likelihood function.
Using a log transformation facilitates easier differentiation, which is crucial when finding maximum likelihood estimates.
In our context of a dataset from a Poisson distribution, we have:

• The log-likelihood function is: \( \ell(\mu) = -(n \mu) + (x_1 + x_2 + \cdots + x_n)\ln(\mu) - \ln(x_1! \cdots x_n!) \).
  
• This function is derived by taking the natural logarithm of the likelihood function.

The benefit of using the log-likelihood is mathematical convenience. It transforms a complex multiplication operation in the likelihood into a simpler addition operation in the log-likelihood. Simplifying things this way makes calculus operations manageable, which is essential for parameter estimation by maximizing the likelihood.