Question

Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a Poisson distribution with parameter \(\mu=1\) (a) Show that the mgf of \(Y_{n}=\sqrt{n}\left(\bar{X}_{n}-\mu\right) / \sigma=\sqrt{n}\left(\bar{X}_{n}-1\right)\) is given by \(\exp \left[-t \sqrt{n}+n\left(e^{t / \sqrt{n}}-1\right)\right]\) (b) Investigate the limiting distribution of \(Y_{n}\) as \(n \rightarrow \infty\). Hint: Replace, by its MacLaurin's series, the expression \(e^{t / \sqrt{n}}\), which is in the exponent of the mgf of \(Y_{n}\).

Short answer

The moment generating function (mgf) of \(Y_{n}\) is given by \(\exp \left[-t \sqrt{n}+n\left(e^{t / \sqrt{n}}-1\right)\right]\). The limiting distribution of \(Y_{n}\), as \(n\) approaches infinity, is the standard normal distribution.

Step-by-step solution

Find the mgf of \(\bar{X}_{n}\)

Given that the sample size is \(n\) and each observation follows a Poisson distribution with parameter \(\mu=1\), the moment generating function \(M(t)\) of a single random variable following a Poisson distribution is given by \(M(t) = e^{\mu(e^t - 1)}\). When calculating the sum of independent variables, this becomes \(M_{\sum X_{i}}(t) = (M_{X_i}(t))^n\), which result in \(M_{\bar{X}_{n}}(t) = e^{n(e^t - 1)}\), because the sum of \(n\) observations divided by \(n\) is equal to the mean of the observations.

Find the mgf of \(Y_{n}\)

Given that \(Y_{n}=\sqrt{n}\left(\bar{X}_{n}-1\right)\), we can find the moment generating function \(M_{Y}(t)\) by substituting \(\bar{X}_{n}\) with \(Y_{n}/\sqrt{n} + 1\) in \(M_{\bar{X}_{n}}(t)\). Hence, we will have \(M_{Y_{n}}(t) = \exp \left[-t\sqrt{n} + n\left(e^{t/\sqrt{n}} - 1\right)\right]\), as required.

Find the first term of the Taylor series for \(e^{x}\)

The Taylor series expansion of \(e^{x}\) around \(x = 0\) is given by \(e^{x} = 1 + x + O(x^2)\). Replacing \(x\) with \(t/\sqrt{n}\), we obtain \(e^{t/\sqrt{n}} = 1 + t/\sqrt{n} + O(1/n)\).

Find the limiting distribution of \(Y_{n}\)

Substituting the Taylor approximation back into the moment generating function for \(Y_{n}\) and taking the limit as \(n\) approaches infinity, we can simplify \(M_{Y_{n}}(t) = \lim_{n \to \infty} \exp\left[-t\sqrt{n} + n\left(1 + t/\sqrt{n} - 1\right)\right] = e^{t^2/2}\). So, \(Y_{n}\) converges in distribution to a normal distribution with mean 0 and variance 1.

Key concepts

Poisson Distribution

The Poisson distribution is a fundamental probability distribution often used to model the number of events that occur in a fixed interval of time or space. Each of these events must be independent, and they typically happen at a constant average rate. The distribution is defined by a single parameter, \( \mu \), which represents the average number of occurrences in that interval.

The probability of observing \( k \) events in an interval given a Poisson distribution is determined by the probability mass function:

• \( P(X = k) = \frac{\mu^k e^{-\mu}}{k!} \)

Here:

• \( \mu \) is the mean number of events.
• \( e \) is Euler's number, approximately equal to 2.71828.
• \( k! \) denotes the factorial of \( k \).

The Poisson distribution simplifies many complex problems, particularly where the main focus is the occurrence of rare events, like the arrival rate of calls at a call center or the occurrence of specific genetic mutations.

The moment-generating function (mgf) for the Poisson distribution is particularly useful as it allows researchers to work with functions of random variables. For a Poisson-distributed random variable with parameter \( \mu \), the mgf is given by:

• \( M(t) = e^{\mu(e^t - 1)} \)

This function simplifies calculations related to problems involving sums of independent Poisson variables.

Limiting Distribution

Limiting distribution refers to the probability distribution to which a sequence of random variables converges as the number of trials, \( n \), approaches infinity. In many statistical convergence problems, understanding the limiting distribution helps in approximating the behavior of large sample sizes through simpler distributions.

In our example, we focus on \( Y_n = \sqrt{n}(\bar{X}_n - 1) \), where \( \bar{X}_n \) is the mean of a random sample from a Poisson distribution. As \( n \to \infty \), we wish to know what kind of distribution \( Y_n \) might approach. Using the Central Limit Theorem, one expects that under certain conditions, the sum of a large number of independent random variables tends to be normally distributed.

For \( Y_n \), by calculating its moment-generating function and using a Taylor Series approximation, we see that as \( n \to \infty \), it approximates a normal distribution with mean 0 and variance 1. This is typically referred to as the standard normal distribution. The moment-generating function for a normal distribution is:

• \( M(t) = e^{\frac{t^2}{2}} \)

This shows how the distribution is symmetric around zero and allows simplified applications in many statistical derivations and analyses.

Taylor Series Expansion

The Taylor series is a powerful mathematical tool used to approximate functions. Consider the function \( e^x \). Its Taylor series expansion around the point \( x = 0 \) is given by:

• \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

This series representation comes in handy in numerous mathematical contexts, specifically for simplifying complex expressions, making calculations manageable.

In this exercise, the Maclaurin series, which is a type of Taylor series, is used to approximate the term \( e^{\frac{t}{\sqrt{n}}} \). When expanded, it becomes:

• \( e^{\frac{t}{\sqrt{n}}} = 1 + \frac{t}{\sqrt{n}} + \frac{t^2}{2n} + O\left(\frac{1}{n^{3/2}}\right) \)

Where \( O\left(\frac{1}{n^{3/2}}\right) \) refers to higher-order terms that become insignificant as \( n \) grows.

This approximation simplifies the moment-generating function for \( Y_n \) and subsequently helps in identifying the limiting distribution. The process of using Taylor series expansion is crucial in deriving results that involve small perturbations around a particular parameter or variable, enabling clearer insights into problem-solving and theoretical analysis.