Statistics: Comprehensive Notes on the Poisson Distribution

Overview of the Poisson Distribution

• The Poisson distribution, along with the Binomial distribution, is one of two specifically named probability distributions discussed in this course.

• It is used to describe specific situations involving the counting of events over a set interval.

• Etymology and History:

  • The word "Poisson" means "fish" in French.

  • The distribution is named after the French mathematician Simeon Poisson, who first described it.

  • Historical Application: One of the first famous applications of this distribution was modeling the number of times soldiers were kicked by donkeys during World War I.

Defining Characteristics of a Poisson Process

• Definition of Successes: A Poisson process involves counting the number of successes within a specified time or space interval.

• Countability: Unlike the Binomial distribution, which has a fixed number of trials ($n$), the Poisson distribution has no defined maximum number of successes. The number of successes can range from $0$ to $\text{infinity}$. This is sometimes referred to as being uncountable in terms of the upper limit.

• Independence: The number of successes are counted in non-overlapping intervals. Therefore, occurrences in one interval are independent of occurrences in another.

• Uniformity across Intervals: The probability of success in any given interval is the same for all intervals of equal size.

• Proportionality: The probability of success is proportional to the size of the interval (e.g., doubling the time interval doubles the expected number of events).

Real-World Examples

• Traffic Accidents: The number of accidents occurring at a specific intersection or traffic light within a set time frame (e.g., between $9:00\text{ AM}$ and $10:00\text{ AM}$).

• Digital Applications: The number of customers using a new banking application within a specific $24\text{-hour}$ period.

• Communications: The number of spam emails received by a user in a given month ($12\text{-month interval}$).

• Manufacturing Defects: The number of defects found in a specified length of material (e.g., a $50\text{-yard}$ roll of fabric).

• Biology/Microbiology: The number of bacteria present in a specific culture or well-defined space, such as a Petri dish.

Mathematical Components and Formulas

• The Random Variable: The Poisson random variable is defined as $X$.

• Probability Mass Function (PMF): The probability of exactly $x$ successes over a given interval is calculated as:     $P(X = x) = \frac{e^{-\mu} \mu^x}{x!}$

  • $x$: The specific number of successes ($x = 0, 1, 2, \dots$).

  • $\mu$ (mu): The mean number of successes over the specified interval. This is typically given or calculated beforehand.

  • $e$: A mathematical constant approximately equal to $2.718$. It is the base of the natural logarithmic scale (often denoted as Ln on calculators, from the French logarithme naturel).

  • $x!$ (Factorial): Calculated as $x \times (x-1) \times (x-2) \times \dots \times 1$.

Measures of Central Tendency and Dispersion

• In a Poisson distribution, the mean and the variance are numerically identical.

• Mean ($\mu$): $\text{Mean} = \mu$

• Variance ($\sigma^2$): $\text{Variance} = \mu$

• Standard Deviation ($\sigma$): To find the standard deviation, take the square root of the variance.     $\sigma = \sqrt{\mu}$

Calculating Poisson Probabilities in Excel

Excel uses the function POISSON.DIST to calculate probabilities. The syntax is POISSON.DIST(x, mean, cumulative).

• Exactly $x$ successes: $P(X = x)$

  • Excel formula: POISSON.DIST(x, mu, 0) (0 or FALSE indicates non-cumulative).

• No more than $x$ successes: $P(X \leq x)$

  • Excel formula: POISSON.DIST(x, mu, 1) (1 or TRUE indicates cumulative).

• At least $x$ successes: $P(X \geq x)$

  • This requires the complement rule: 1 - P(X < x), which for discrete values is $1 - P(X \leq x - 1)$.

  • Excel formula: 1 - POISSON.DIST(x - 1, mu, 1).

Case Study: Craft Brewery Openings

Scenario Data

• Observed Rate: $1.5$ craft breweries open every day in the USA.

• Interval required for questions: One week ($7\text{ days}$).

Adjusting the Mean ($\mu$)

Because the question asks for weekly probabilities but provides daily data, the mean must be adjusted to match the time interval: $\mu = 1.5\text{ breweries/day} \times 7\text{ days/week} = 10.5\text{ breweries/week}$

Example Calculations

1. Probability that no more than 10 breweries open in a week ($P(X \leq 10)$):

   • Formula: POISSON.DIST(10, 10.5, 1)

   • Result: Approximately $0.52$ (or $52\%$).

2. Probability that exactly 10 breweries open in a week ($P(X = 10)$):

   • Formula: POISSON.DIST(10, 10.5, 0)

   • Result: Approximately $0.12$ (or $12\%$).

3. Probability that at least 10 breweries open in a week ($P(X \geq 10)$):

   • Formula: 1 - POISSON.DIST(9, 10.5, 1)

   • Context: Since $10$ is included in the "at least" range, the complement must include only values below $10$ (i.e., $\leq 9$).

   • Result: Approximately $0.60$ (or $60\%$).

4. Statistical Parameters for the week:

   • Mean: $10.5$

   • Variance: $10.5$

   • Standard Deviation: $\sqrt{10.5} \approx 3.24$

Questions & Discussion

• Question: How do we handle changes in units of time in Poisson problems?

  • Response: Always look at the unit of time in the final question. If you are given a daily rate but asked for a weekly probability, you must scale the mean ($\mu$) accordingly by multiplying the daily rate by $7$.

• Anecdote: The instructor mentioned that the word "breweries" is difficult to pronounce, comparing it to the fictional show "The Rural Juror" from the television series 30 Rock.

• Exam Tip: The instructor emphasized that keeping these Excel formulas handy will be very helpful for the next exam.