Question

Using the computer, obtain an overlay plot of the pmfs of the following two distributions: (a) Poisson distribution with \(\lambda=2\). (b) Binomial distribution with \(n=100\) and \(p=0.02\). Why would these distributions be approximately the same? Discuss.

Short answer

The given Poisson and Binomial distributions appear similar as the product \( np \) is equal to \( \lambda \), a characteristic condition for a Binomial distribution to approximate to a Poisson distribution, especially when \( n \) is large and \( p \) is small.

Step-by-step solution

Plot the Poisson Distribution

Using a software like Python or MATLAB, code a Poisson Distribution with \( \lambda = 2 \). This distribution models the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Plot the Binomial Distribution

In the same overlay, write the code for a Binomial Distribution with \( n = 100 \) and \( p = 0.02 \), using the same software as before. This distribution models the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.

Compare and Discuss

Upon overlay, it would be observed that the plotted distributions are almost similar. This occurs because of the relationship between the Poisson and Binomial distributions. When \( n \) is large and \( p \) is small, such that \( np = \lambda \), the binomial distribution approximates to a Poisson distribution. This approximation is usually effective when \( p \) is less than 0.05 and \( n \) is at least 20. In this instance, \( np = 2 \) showing that the approximation is effectively equal to the Poisson distribution parameter \( \lambda = 2 \).

Key concepts

Poisson Distribution

The Poisson distribution is a probability model frequently used in situations where we are counting the number of times a certain event occurs within a defined space or time interval. For example, it can represent the number of emails a person receives per day or the number of meteors hitting the earth per year.

Mathematically, it is defined by its parameter, lambda (\( \boldsymbol{\textlambda} \), which signifies the average rate at which events occur. The probability mass function (pmf) of the Poisson distribution gives us the probabilities of observing various counts of the event. It assumes that events occur randomly and independently of each other, with no two events happening at the same instant.

Binomial Distribution

On the other hand, the Binomial distribution applies to scenarios where we conduct a fixed number of independent experiments, and each has a binary outcome: success or failure. Common examples include flipping a coin certain times and counting the heads, or checking for defective items in a batch of products.

The parameters of a binomial distribution are the number of trials, denoted by n, and the probability of success in each trial, denoted by p. Its probability mass function (pmf) tells us the likelihood of achieving a certain number of successes out of n trials. The key characteristics of this distribution are the independence of trials and the constant probability of success.

Probability Mass Function (pmf)

A probability mass function (pmf) serves as a tool to express the probabilities associated with a discrete random variable. It's essentially a function that maps any possible value of a discrete random variable to its probability.

For Poisson:

The pmf is denoted by: \[ P(X = k) = \frac{e^{-\textlambda} \textlambda^k}{k!}\] for k = 0, 1, 2, ...

For Binomial:

It is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \[ \binom{n}{k} \] is the binomial coefficient, representing the number of ways to choose k successes from n trials.

The pmf enables us to predict the likelihood of various outcomes, which is fundamental in the study of probability and statistics.

Distribution Approximation

Distribution approximation is a technique used in statistics to find a simpler distribution that closely mirrors the properties of a more complex distribution, given certain conditions. The power of this technique lies in its ability to simplify complex probability problems.

In the context of the exercise, the Binomial distribution can be approximated by the Poisson distribution when the number of trials (n) is large, and the probability of success (p) is small, provided that the product of n and p (\( np \) is roughly equal to the mean rate (\( \textlambda \) of the Poisson distribution. This situation is especially true when p is less than 0.05 and n is greater than 20. The key takeaway is that while these distributions are derived from different scenarios, under specific conditions, they can be quite similar and used interchangeably for practical purposes.