Bernoulli Binomial Updated Fall 2024

Discrete Distributions

Overview

• Discrete distributions describe the probability of outcomes in scenarios where the set of possible results is countable.

Types

• Examples of Discrete Probability Distributions:

  • Binomial

  • Poisson

  • Bernoulli

  • Uniform

Continuous Probability Distributions

Overview

• Continuous distributions describe the probability of outcomes in scenarios where the outcomes can take on any value in a range.

Bernoulli Trials

• Definition: A sequence of experiments with two possible outcomes.

• Examples of Applications:

  • Wins in successive plays of a game

  • Incidence of disease

  • Servers down in a server farm

  • Gambling applications

  • Political polls (where each trial is a surveyed opinion)

Independent Trials

• Key Concept: If events are independent, the probability of multiple events occurring together is the product of their individual probabilities.

• Example Calculation:

  • Given a probability of 0.02 for at least one defective part on an assembly line, the probability of observing 5 consecutive zero defect minutes is = 0.98^5 = 0.904.

Bernoulli Distribution

• Key Characteristics:

  • Two mutually exhaustive outcomes: SUCCESS (P(success)=𝜋) and FAILURE (P(failure)=1-𝜋).

  • Random variable X is defined as:

    • X=1 (success)

    • X=0 (failure)

  • Probability function:

    • P(0) = 1 - 𝜋

    • P(1) = 𝜋

  • Notation: X ~ Bernoulli(𝜋)

Mean and Variance of Bernoulli Random Variable

• Mean (𝜇𝑋): E(X) = 𝜋

  • Interpreted as the share of successes in a population.

• Variance (𝜎𝑋²):

  • Variance formula: 𝜎𝑋² = 𝜋(1 - 𝜋)

  • Represents variability around the mean success rate.

Binomial Probability Distribution

• Definition: A generalization of the Bernoulli distribution, considering multiple independent trials.

• Example: Tossing a coin multiple times.

  • Let n = number of independent trials; π = probability of success in a single trial; X = number of successes.

Example Scenarios:

1. Tossing a Coin Once:

   • Sample space: {H, T}

   • Bernoulli distribution.

2. Tossing a Coin Twice:

   • Sample space expands to {HH, HT, TH, TT}, evaluating P(heads) in combinations.

   • Result is represented by Binomial random variable.

Probabilities Calculation for Coin Tossing

• General Approach:

  • For example, when tossing a coin three times, probabilities for number of heads can be computed, i.e., P(X=0), P(X=1), etc.

• Formulae for computations include:

  • P heads = c * (1/2)^n for n tosses where c is combinations of favorable outcomes for heads.

Characteristics of the Binomial Distribution

1. Independent Observations: Each trial does not influence others.

2. Fixed Probability: Probability of success is constant for each trial.

3. Event Categories: Outcomes classified as success or failure.

Applications of the Binomial Distribution

• Used in various fields such as:

  • Manufacturing (detecting defective items)

  • Marketing research (customer purchase behavior)

  • Statistical sampling (customer churn tracking)

Probability Distribution Function for Binomial Random Variable

• Mathematical representation:

  • P(X = x | n, π) = (n!)/(x!(n-x)!) * (π^x) * ((1-π)^(n-x))

  • Describes the probability of x successes in n trials.

Characteristics of the Binomial Distribution

• Mean: μ = n * π

• Variance: σ² = n * π * (1 - π)

Examples of Probability Calculations Using Binomial Distribution

1. Calculation for Defective Computers:

   • Given π=0.02 for faulty computers, the probability for observed defective computers in tests follows the binomial distribution model.

2. Excel Calculation for Binomial Distribution:

   • For practical applications, functions like BINOM.DIST can compute probabilities directly.