Poisson and Hypergeometric Distributions: Lecture Notes and Assessment Guidelines

Administrative Announcements and Assessment A1 Details

• Official Communication Channel: The announcements forum is the primary source for binding information regarding assessments and course changes. It is separate from the general use forum. Students are expected to check it regularly, as answers to common email queries are typically found there.

• Assessment A1 Content: The specific chapters and topics covered in the A1 assessment were detailed in the announcement posted on March 12.

• Test Logistics:

  • Date and Window: The quiz is scheduled for next Thursday and will be open from 5:00 PM to 9:00 PM.

  • Duration: Once a student starts the quiz, they have 2 hours to complete it.

  • Lateness Warning: Students must start the quiz by 7:00 PM at the latest to receive the full 2-hour duration. If started later (e.g., 8:00 PM), the quiz will still close automatically at 9:00 PM.

• Testing Format and Software:

  • Structure: The test is a mix of an application and a Tutorial (TUT) style. Students will download an Excel dataset and toggle between Excel and the quiz interface.

  • Marking: Only the final answers entered into the quiz are graded. Work done in the Excel document is not marked.

  • Computation: Manual calculation is technically allowed where possible, but several tasks (such as generating specific graphs from previous tutorials) require Excel.

  • Software Readiness: Students using campus computer areas or personal laptops must ensure the "Data Analysis" add-in (used for histograms) and a stable internet connection are functional before opening the quiz.

• Revision Schedule:

  • Lectures: Next Monday and Wednesday will be dedicated solely to A1 revision. No new material will be introduced.

  • Tutorial (TUT): Revision will also take place in the tutorial. The Thursday session from 3:00 PM to 5:00 PM is typically the most full.

Characteristics and Assumptions of the Poisson Distribution

• Definition: The Poisson distribution measures the probability of a discrete number of events occurring over a continuous area of opportunity (typically an interval of time).

• Area of Opportunity: This is a defined, uniform interval (e.g., one minute, one hour, or a specific space). Consistently sized intervals are required for accurate measurement.

• Key Assumptions:

  • Constant Probability: The probability that an event occurs in a given area of opportunity must be the same for all areas of opportunity. There should be no irregular spikes or dips in probability.

  • Independence: The occurrence of one event must not influence the probability of another event occurring (events are independent).

  • Coherence: As the area of opportunity increases, the expected number of events increases; conversely, if the interval decreases, the probability of an event happening decreases.

• Parameter: The distribution is characterized by a single parameter, lambda ($\lambda$), which represents the average rate or mean number of events in a specified area of opportunity.

Mathematical Formulation of the Poisson Distribution

• The Probability Formula: To calculate the probability of a specific number of events ($x$) occurring, the formula is:          $P(x | \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}$     

  • $e$: A mathematical constant approximately equal to 2.71828. On a Casio calculator, it is typically accessed via SHIFT + ln.

  • $\lambda$: The average number of events in the specified interval.

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

  • $x!$: The factorial of $x$.

Practical Examples and Unit Conversion in Poisson Distributions

• Case Study: Bank Customer Arrivals:

  • Scenario: Customers arrive at a rate of 3 per minute during the noon hour.

  • Given: $\lambda = 3\, \text{customers/minute}$.

  • Task: Find the probability that exactly 2 customers arrive in a specific minute.

  • Calculation:                  $P(x = 2 | \lambda = 3) = \frac{e^{-3} 3^2}{2!}$         

  • Result: $0.2240$.

• Handling Mismatched Units (The Curveball):

  • Scenario: Using the same bank data where $\lambda = 3\, \text{per minute}$, find the probability that 10 customers arrive in a given hour.

  • Method: Before applying the formula, $x$ and $\lambda$ must share the same units of time. Since the goal involves an hour, $\lambda$ must be converted.

  • Unit Conversion: $\lambda = 3\, \text{per minute} \times 60\, \text{minutes} = 180\, \text{per hour}$.

  • Calculation:                  $P(x = 10 | \lambda = 180) = \frac{e^{-180} 180^{10}}{10!}$

Statistical Properties and Visualizing the Poisson Distribution in Excel

• Descriptive Statistics for Poisson:

  • Mean ($\mu$): For this distribution, the expected value is simply $\lambda$.

  • Variance ($\sigma^2$): Uniquely, the variance is also equal to $\lambda$.

  • Standard Deviation ($\sigma$): Calculated as $\sqrt{\lambda}$.

• Relationship between Rate and Spread: As the average rate of occurrence ($\lambda$) increases, the randomness and spread (variance) of the distribution also increase.

• Excel Implementation: The function POISSON.DIST(x, mean, cumulative) is used.

  • x: Number of events.

  • mean: Lambda ($\lambda$).

  • cumulative: Use FALSE for exact probability ($P(X = x)$) and TRUE for cumulative probability ($P(X \le x)$).

• Graphical Representation:

  • When $\lambda$ is small (e.g., 3), the distribution is right-skewed and concentrated near zero.

  • As $\lambda$ increases (e.g., shifting from 10 to 20 to 50), the distribution moves to the right and becomes more symmetrical, while the spread becomes noticeably wider.

  • Unlike the Binomial distribution, the Poisson has no theoretical upper limit for $x$, although the probability for extremely high values eventually becomes infinitesimal (approaching zero in scientific notation).

Introduction to the Hypergeometric Distribution

• Comparison with Binomial Distribution:

  • Binomial: Assumes sampling with replacement or sampling from an infinite population. Trials are independent.

  • Hypergeometric: Used when sampling without replacement from a finite population. Because items are not replaced, the probability of selecting an item changes with each trial, making the trials dependent.

• Population Layout:

  • $N$: Total population size.

  • $e$: Total number of items of interest (successes) in the population.

  • $n$: Size of the sample taken.

  • $x$: Number of items of interest found within the sample.

Mathematical Formulation and Example of the Hypergeometric Distribution

• The Probability Formula:          $P(x | n, e, N) = \frac{\binom{e}{x} \binom{N-e}{n-x}}{\binom{N}{n}}$     

  • $\binom{e}{x}$: Number of ways to select $x$ items of interest from $e$.

  • $\binom{N-e}{n-x}$: Number of ways to select the remaining non-interest items.

  • $\binom{N}{n}$: Total number of ways to select a sample of size $n$ from population $N$.

• Example: Checking for Illegal Software:

  • Scenario: 10 computers out of a department of 30 have illegal software. 8 computers are checked out at random.

  • Parameters: $N = 30$, $e = 10$, $n = 8$, and we want the probability that $x = 2$.

  • Calculation:                  $P(x = 2) = \frac{\binom{10}{2} \binom{30-10}{8-2}}{\binom{30}{8}} = \frac{\binom{10}{2} \binom{20}{6}}{\binom{30}{8}}$         

  • Calculator Tip: Use the fraction button followed by the combination function (e.g., 10 SHIFT nCr 2).

  • Result: $0.2980$ (approximately 29.8%).

Statistical Properties of the Hypergeometric Distribution

• Expected Value (Mean):          $\mu = n \times \frac{e}{N}$     

• Standard Deviation: The standard deviation includes the Finite Population Correction (FPC) factor to account for sampling without replacement:          $\sigma = \sqrt{\frac{n \times e \times (N-e)}{n^2}} \times \sqrt{\frac{N-n}{N-1}}$     

  • Note: While this exact detail may not be immediate for the upcoming test, the correction factor is fundamental for dealing with finite populations.