Poisson Random Variable (Poisson RV)

Introduction to the Poisson Random Variable

The Poisson random variable models the number of times a discrete phenomenon occurs in a fixed interval of time or space, given that these occurrences are independent and happen at a constant average rate. This distribution is widely used in statistics to model rare events.

• Key characteristics: Events occur independently and at a constant average rate (λ).

• Examples of phenomena:

  • Number of tweets posted per second

  • Number of car crashes in a day

  • Number of times a person sneezes in a week

  • Number of questions asked by students during a lecture

Definition and Properties

• Notation: If X is a Poisson random variable with rate λ, we write X ~ Pois(λ).

• Support: The possible values of X are the non-negative integers: {0, 1, 2, ...}.

• Probability Mass Function (PMF):

• Mean (Expected Value):

• Variance:

• Standard Deviation:

Example 1: Customer Arrivals

Suppose a new store in La Jolla receives customers 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 the rate for 30 minutes: (since 30 minutes is three 10-minute intervals).

• Step 2: Use the PMF:

Example (at least 4 customers): To find the probability of at least 4 customers, sum the probabilities for 4, 5, 6, ... or use .

Example 2: Fatal Car Accidents

Suppose fatal car accidents in a town occur at a rate of 1.5 per month. What is the probability of at most one fatal accident in the next two months?

• Step 1: For two months, .

• Step 2: , where

Expected number of accidents in two months:

Probability of at least 3 accidents in one month: for

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 .

• Let X be the number of points scored in one game:

• Probability of scoring 0 points:

• Expected number of games with 0 points:

• Probability of scoring 1 point:

• Expected number of games with 1 point:

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

Poisson Approximation to the Binomial

When to Use the Poisson Approximation

When the number of trials n is large and the probability of success p is small, the Binomial distribution can be approximated by a Poisson distribution with .

• Example: If 1 out of every 5000 items is defective, use the Poisson approximation to estimate the probability of finding at least 2 defective items in a batch of 7000.

• Example: For a rare disease affecting 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.

Recap: Discrete Random Variables (DRVs)

Types of 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, ...}.

• 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, ..., 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 λ, support is {0, 1, 2, ...}.