Bus. Stats - Feb 20 Recording

Discrete Probability Distributions

Overview

• Focus on Poisson and Hypergeometric distributions.

• Assignments and solutions available on Canvas; classwork assistance provided.

Poisson Distribution

• Definition: Used for modeling count data (e.g., number of successes in a time segment).

• Examples:

  • Scholarship offers for high school athletes from universities.

  • Number of emergency calls in an hour at a hospital.

  • Potholes in a specified highway segment.

Key Characteristics

• Count data consists of whole numbers and cannot be fractional (e.g., 1 or 2 scholarships).

• Assumption: Average number of successes in one segment (denoted by lambda, λ).

• Trials are independent, with consistent probability of success across segments.

Application Example

• Scenario: Average bank customers arriving is 16 per hour (λ = 16).

• Calculating Probabilities:

  • Use formula: P(x) = (λ * t)^x * e^(-λ * t) / x!

  • Where:

    • λ = average number of successes

    • t = time segment

    • x = number of successes

Practical Calculation Using Excel

• Setup: Input average, time, and number of successes in designated cells.

• Formula for Exact Probability: =POISSON.DIST(x, λ*t, FALSE) for exact occurrences; =POISSON.DIST(x, λ*t, TRUE) for cumulative probability.

Assumptions and Limitations

• Both expected value and variance in Poisson distribution are equal to λ * t.

• Assumes mean and variance equality, which often doesn’t hold in real scenarios.

• Poisson is less practical; consider using the negative binomial distribution instead for systems where they differ.

Hypergeometric Distribution

• Usage: Best for dependent trials where the probability of success changes from trial to trial.

• Overview:

  • Used for scenarios where items are chosen without replacement.

  • Important in binary outcomes but with dependent trials.

Hypergeometric Formula

• Formula: P(X = x) = [C(N - K, n - k) * C(K, k)] / C(N, n)

  • Where:

    • N = population size

    • K = total number of successes in the population

    • n = sample size

    • k = number of observed successes in the sample

Example Scenario

• Context: Firm downsizing with a set population of employees (30 total).

• Layoffs randomly select 10 employees; success defined as female staff members.

• Calculate probabilities for different outcomes (e.g., eight women laid off).

Excel Calculations for Hypergeometric Distribution

• Set required population values and sample sizes in Excel for easy probability calculations.

• Use formula in Excel to streamline calculations without needing to compute manually.

Comparison of Distributions

• Binomial Distribution: Applicable for independent trials with binary data.

• Poisson Distribution: For independent trials with count data.

• Hypergeometric Distribution: For dependent trials with either count or binary data.

• Distinguish use cases based on the independence/dependency of trials and nature of the data.