Poisson Random Variable and Related Discrete Distributions

Poisson Random Variable (Poisson RV)

Definition and Context

The Poisson random variable is a discrete random variable that models the number of times a certain event occurs in a fixed interval of time or space, given that these events happen independently and at a constant average rate. This distribution is widely used in situations where events occur randomly and independently over time or space.

• Examples of phenomena:

  • Number of tweets posted on Twitter in a second

  • Number of car crashes in the world in a day

  • Number of times a person sneezes in a week

  • Number of questions a student asks during a lecture

• Parameter: λ (lambda) is the average rate of occurrence in the given time frame.

Probability Model and Properties

• Notation: $X \sim \text{Pois}(\lambda)$

• Support: $X \rightarrow \{0,1,2,3,\ldots\}$

• Probability Mass Function (PMF):

$P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}, \quad x = 0,1,2,\ldots$

• Mean: $E(X) = \lambda$

• Variance: $\text{Var}(X) = \lambda$

• Standard deviation: $sd(X) = \sqrt{\lambda}$

Worked Examples

Example 1: Customer Arrivals

Suppose a new store in La Jolla has customers arriving at a rate of 1 customer every 10 minutes. What is the probability of getting 4 customers in the next 30 minutes?

• Step 1: Determine $\lambda$ for 30 minutes: $\lambda = 3$ (since 30 minutes is three 10-minute intervals).

• Step 2: Use the PMF: $P(X = 4) = \frac{e^{-3} 3^4}{4!}$

Variation: What is the probability of getting at least 4 customers in the next 30 minutes?

• Calculate $P(X \geq 4) = 1 - P(X \leq 3)$, where $P(X \leq 3)$ is the sum of probabilities for $X = 0,1,2,3$.

Example 2: Fatal Car Accidents

Suppose fatal car accidents in a town happen at a rate of 1.5 per month.

• At most one fatal accident in two months: $\lambda = 3$ (since 2 months × 1.5/month)

• Calculate $P(X \leq 1) = P(X=0) + P(X=1)$ using the PMF.

• Expected number of fatal accidents in two months: $E(X) = \lambda = 3$

• Probability of at least 3 accidents in one month: $\lambda = 1.5$, $P(X \geq 3) = 1 - P(X \leq 2)$

Poisson Model in Real Life

Application: Wayne Gretzky's Hockey Points

Wayne Gretzky scored 1669 points in 696 games. The average number of points per game is $\lambda = 1669/696 \approx 2.398$.

• Let $X$ be the number of points scored in one game: $X \sim \text{Pois}(2.398)$

• Probability of scoring 0 points: $P(X=0) = e^{-2.398} \approx 0.091$

• Expected number of games with 0 points: $696 \times 0.091 \approx 63.3$

• Probability of scoring 1 point: $P(X=1) = e^{-2.398} \frac{2.398^1}{1!} \approx 0.218$

• Expected number of games with 1 point: $696 \times 0.218 \approx 151.7$

Points	Games	Expected
0	69	63.3
1	155	151.7
2	171	181.9
3	143	145.4
4	79	87.1
5	57	41.8
6	14	16.7
7	6	5.7
8	2	1.7
9 plus	0	0.6

Interpretation: The observed and expected values are in good agreement, showing that the Poisson model is appropriate for this type of data.

Poisson Approximation to the Binomial

When to Use

When the number of trials $n$ is large and the probability of success $p$ is small, the Binomial distribution $\text{Bin}(n, p)$ can be approximated by a Poisson distribution with $\lambda = np$.

• Example 1: If one item out of every 5000 is defective, use the Poisson approximation to estimate the probability of finding at least five defective items in a batch of 7000.

• Example 2: If a rare disease affects 1 in 100,000 people, in a university of 35,821 students, use the Poisson approximation to compute the probability that no one has the disease.

Formula: For $n$ large, $p$ small, $X \sim \text{Bin}(n, p) \approx \text{Pois}(\lambda)$ where $\lambda = np$.

Recap: Discrete Random Variables (DRVs)

Summary Table of Key Discrete Random Variables

• Geometric random variable: Counts the number of trials needed until the first success.

  • Example: Number of coin tosses until the first head.

  • Key points: Requires Bernoulli trials, probability of success $p$, support is $\{1,2,3,\ldots\}$.

• Binomial random variable: Counts the number of successes in a fixed number of trials.

  • Example: Number of heads in 10 coin tosses.

  • Key points: Requires Bernoulli trials, probability of success $p$, fixed number of trials $n$, support is $\{0,1,2,\ldots,n\}$.

• Poisson random variable: Counts the number of times a phenomenon occurs in a fixed time frame.

  • Example: Number of phone calls received in one hour.

  • Key points: Requires average rate $\lambda$, support is $\{0,1,2,\ldots\}$.