DISCRETE DISTRIBUTIONS

  • Discrete random variables have finite or countably infinite values.

  • Key discrete distributions: Binomial, Poisson, Hypergeometric.

BINOMIAL DISTRIBUTION INTRODUCTION

  • Focuses on binomial distribution applications.

  • Preliminary discussion on uniform and Bernoulli distributions.

BERNOUILLI DISTRIBUTION

  • A trial with two outcomes (success = 1, failure = 0).

  • Examples include coin flips and target hits.

BINOMIAL PROBABILITY DISTRIBUTION

  • Generalization of Bernoulli trials to n independent trials.

  • Coin-tossing as a basic binomial example.

BINOMIAL EXPERIMENT CHARACTERISTICS

  1. Fixed number of trials (n).

  2. Two possible outcomes (success or failure).

  3. Constant probability of success (p) across trials.

  4. Independent trials.

  5. Focus on the number of successes (x).

NON-BINOMIAL EXAMPLE

  • Drawing cards without replacement does not meet criteria (changing probabilities, dependent trials).

BINOMIAL PROBABILITY FORMULA

  • For n trials with success probability p, probability of k successes:
    P(k)=(nk)pk(1−p)n−kP(k) = {n \choose k} p^k (1-p)^{n-k} where q = 1-p.

MEAN AND STANDARD DEVIATION

  • Mean: μ=np\mu = np

  • Standard deviation: σ=np(1−p)\sigma = \sqrt{np(1-p)}

EXAMPLE PROBLEM

  • A marksman hits 80% of the time, firing 5 shots:

    • Probability of exactly 3 hits.

    • Cumulative probabilities were discussed.

PRACTICE PROBLEMS

  1. Compute probabilities for varied trials and successes in binomial distributions.

  2. Apply binomial formulas to examples with different sample sizes and success rates.