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:

  1. X is a discrete random variable
  2. X is the number of occurrences of an event over some interval (time, distance, area, volume)
  3. The occurrences must be random — occurrences do not follow any pattern; they are unpredictable
  4. 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.

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