FAD1015 L14 — Poisson Distribution

Lecture 14 covering the Poisson distribution for modeling rare events. Source file: Week 7 L14 Poisson Distribution.pdf

Summary

Study of the Poisson distribution: setting/characteristics, probability mass function, Poisson distribution tables, mean, variance, and use as an approximation to the binomial distribution.

Key Concepts

Probability Distributions — Poisson distribution
Poisson Process — events occurring at constant rate over an interval
Rate Parameter (λ) — average occurrences per interval
Poisson Probability Formula
Mean = Variance = λ
Poisson Approximation to Binomial

Lecture Coverage

1. Poisson Setting / Characteristics

The following conditions must be satisfied to apply the Poisson probability distribution:

X is a discrete random variable
X is the number of occurrences of an event over some interval (time, distance, area, volume)
The occurrences must be random — occurrences do not follow any pattern; they are unpredictable
The occurrences must be independent of each other — one occurrence (or non-occurrence) does not influence successive occurrences

Examples:

Number of patients arriving at the emergency ward during a one-hour interval
Number of defective items in the next 100 items manufactured
Number of accidents on a highway during a one-week period
Number of customers coming to a grocery store during a one-hour interval
Number of television sets sold during a given week

2. Differences from Binomial Distribution

Feature	Binomial	Poisson
Parameters	Sample size $n$ and probability $p$	Mean $\lambda$ only
Possible values	$0, 1, 2, \ldots, n$	$0, 1, 2, \ldots$ (no upper limit)

3. Probability Mass Function

If a random variable $X$ has a Poisson probability distribution with parameter $\lambda$, then:

$$X \sim P_o(\lambda)$$

$$P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!}$$

where:

$x =$ number of occurrences $= 0, 1, 2, \ldots$
$\lambda =$ mean number of occurrences in an interval; $\lambda > 0$
$e \approx 2.71828$

Key properties:

The Poisson distribution deals with the frequency of an event in a specific interval
The probability of an event occurring is proportional to the size of the interval
No upper limit on the number of events

4. Using the Poisson Table

Cumulative probabilities are calculated as:

$$P(X \geq r) = P(X = r) + P(X = r+1) + P(X = r+2) + \ldots + P(X = n)$$

Tables provide cumulative upper-tail probabilities for given $m = \lambda$ and $r$ values.

5. Mean and Variance

If $X \sim P_o(\lambda)$, then:

Mean: $\mu = \lambda$
Variance: $\sigma^2 = \lambda$
Standard Deviation: $\sigma = \sqrt{\lambda}$

The mean and variance of the Poisson distribution are both equal to the parameter $\lambda$ itself.

6. Poisson Approximation to the Binomial Distribution

The binomial distribution tends toward the Poisson distribution when $n \to \infty$, $p \to 0$, and $\lambda = np$ stays constant.

When $n$ is large and $p$ is very small, the binomial distribution can be approximated by a Poisson distribution.

Rule of thumb:

If $n > 20$ and $np < 5$ OR $nq < 5$, then Poisson is a good approximation

New parameter: $\lambda = np$

7. Worked Examples

Example 1 — Using the Formula A car breaks down an average of three times per month. Find the probability that during the next month, this car will have: a) Exactly two breakdowns b) At most one breakdown

Example 2 — Using the Poisson Table For $X \sim P_o(3)$: a) $P(X = 2)$ b) $P(X \leq 1)$

Example 3 — Mean and Variance A car breaks down an average of three times per month. Find the mean and variance of the distribution.

Example 4 — LAZADA Returns LAZADA provides free examination for seven days. On average, 2 of every 10 products sold are returned. Find the probability that: a) Exactly 6 of 40 products sold are returned b) Exactly 3 of 25 products sold are returned c) Less than 5 of 60 products sold are returned d) More than 1 of 5 products sold are returned

Example 5 — Rate Adjustment The number of calls arriving at a switchboard each hour is 180. Determine the probability that in a randomly chosen minute, the number of calls is: a) Less than 6 b) More than 2 c) More than 2, given less than 6 arrivals

Example 6 — Poisson Approximation The probability of any one letter being delivered to the wrong house is 0.03. On a random day Mr Postman delivers 100 letters. Using a Poisson approximation, find the probability that at least 4 letters are delivered to the wrong house.

Example 7 — Poisson Approximation (Large n, small q) Let $X \sim B(60, 0.95)$. Use the Poisson approximation to find: a) $P(X = 50)$ b) $P(X \geq 44)$

8. Exercises

Exercise 1 If $X \sim P_o(1.8)$, find the following probabilities by using the formula and the Poisson distribution table (round to four decimal places): a) $P(X = 1)$ b) $P(X \geq 2)$ c) $P(X < 1)$ d) $P(X \leq 1)$ e) $P(X > 3)$ f) $P(0 < X < 3)$ g) $P(2 \leq X \leq 4)$ h) $P(0 \leq X < 2)$

Exercise 2 Yummy expected to receive 4 emails in a week. Find the probability of receiving: a) No emails this week b) 3 emails at most this week c) 8 emails for the next 2 weeks Use the formula and the Poisson distribution table. Round to four significant figures.

Exercise 3 A rental car service company has 5 cars available each day. The number of cars rented out each day is randomly distributed with a mean of 2. Find the probability that the company cannot meet the demand for cars on any one day. Round to four decimal places.

Exercise 4 The number of accidents at a junction in Jalan Duta, Kuala Lumpur averages four per week. Calculate the probability that the number of accidents is: a) At most one over a period of one week b) Exactly three over a fortnight c) Zero over a period of three weeks d) Exactly five in a month Round to four decimal places.

Exercise 5 If $X \sim P_o(\beta)$ and $P(X = 0) = 0.0005$, find: a) The value of $\beta$ b) $P(X = 7)$

Exercise 6 If $X$ is a discrete random variable with mean $\lambda$ and $P(X = 0) = 0.0302$, find: a) The value of $\lambda$ b) $P(X = 4)$

Exercise 7 $X$ is a discrete Poisson random variable with parameter $\lambda = 3.5$. Find the probabilities: a) $P(X = 2)$ b) $P(X < 3)$ c) $P(X = 2 \mid X < 3)$ Round to four decimal places.

Related Course Page

FAD1015 - Mathematics III