Section4.2-R

Introduction

  • Topic: Binomial Distribution

  • Presenter: Eddy Kwessi, Department of Mathematics, Trinity University

  • Date: October 22, 2024

Goals

  • Study two discrete distributions:

    • Bernoulli Distribution

    • Binomial Distribution

Bernoulli Distribution

  • Definition: A Bernoulli trial has two outcomes: success or failure.

  • Random Variable Representation:

    • Let X = 1 for success and X = 0 for failure.

    • Probabilities: Pr(X = 1) = p, Pr(X = 0) = 1 - p.

  • Notation: X ∼ Bernoulli(p)

  • Probability Mass Function (p.m.f):

    • p(x) = p^x (1 - p)^(1 - x), where x = 0, 1

  • Population Average:

    • µX = p

  • Population Standard Deviation:

    • σX = sqrt(p(1 - p))

Examples of Bernoulli Trials

  1. Coin Flip:

    • Success: Getting a head

    • Pr(X = 1) = 1/2, Pr(X = 0) = 1/2

  2. Die Roll:

    • Success: Getting an ace

    • Pr(X = 1) = 1/6, Pr(X = 0) = 5/6

Binomial Distribution

  • Definition: Conduct n independent Bernoulli trials, success probability p.

  • Random Variable: X = number of successes in n trials.

  • Notation: X ∼ Binom(n, p)

  • p.m.f:

    • p(x) = (n choose x) * p^x * (1 - p)^(n - x), where x = 0, 1, ..., n

  • Population Average:

    • µX = n * p

  • Population Standard Deviation:

    • σX = sqrt(n * p * (1 - p))

Example of Binomial Distribution

  • Rolling a Die:

    • Let X be the number of 6's rolled in 3 trials.

    • Possible values: 0, 1, 2, 3.

    • p.m.f displayed as probability table and general formula.

R-Commands for Binomial Distribution

  1. Calculate Pr(X = k):

    • dbinom(k, n, p)

  2. Calculate Pr(X ≤ k):

    • pbinom(k, n, p)

  3. Calculate Pr(X ≥ k):

    • 1 - pbinom(k - 1, n, p)

Practice Problems

  1. Die Example: Probability of rolling a 6 exactly twice with 8 rolls.

    • Solution involves understanding Binomial distribution parameters and applying p.m.f.

  2. Health Insurance Example: Finding the distribution of insured Americans in a sample.

    • Use probabilities based on specific criteria and R-commands.

Homework

  • Read Section 4.2 (Pages 212-213)

  • Complete Problems: 4, 8, 13, 17, 20 by October 30.