Lecture 4 - Discrete Probability Distributions

DISCRETE PROBABILITY

  • Discrete Random Variables:

    • Types: Binomial, Poisson, Hypergeometric

DISCRETE PROBABILITY VARIABLES

  • Distinction from continuous random variables.

DISCRETE PROBABILITY DISTRIBUTION

  • Defines possible values of a discrete random variable (X) and corresponding probabilities.

  • Format: table, graph, or mathematical formula.

  • Condition: 0 ≤ P(x) ≤ 1; P(X) must sum to 1.

EXAMPLES OF DISCRETE PROBABILITY DISTRIBUTIONS

  • Example 1: Valid distributions based on given probabilities; identify valid setups.

MEAN OF A DISCRETE RANDOM VARIABLE

  • Formula: Mean (𝜇) = Σ[x * P(x)]

  • As sample size increases, mean converges toward expected value.

STANDARD DEVIATION OF A DISCRETE RANDOM VARIABLE

  • Formula: σ = √(Σ[(x - μ)² * P(x)])

  • Represents variability in the outcomes.

DISCRETE UNIFORM DISTRIBUTION

  • Equal probabilities across outcomes; example using light bulbs.

  • Mean and standard deviation formulas for this distribution.

BERNOULLI PROCESS

  • Repeated trials with binary outcomes (success or failure).

  • Key properties: Constant probability of success, independence of trials.

BINOMIAL PROBABILITY DISTRIBUTIONS

  • Characteristics: Fixed number of trials, independent trials, two outcomes (success/failure).

  • Probability function used for calculating outcomes in a binomial experiment.

POISSON PROBABILITY DISTRIBUTION

  • Ideal for counting occurrences of events in fixed intervals.

  • Conditions: Independence, constant rate of occurrence.

HYPERGEOMETRIC PROBABILITY DISTRIBUTION

  • Used when sampling without replacement from a finite population.

  • Characteristics of the population and sample size impact the distribution.

MULTINOMIAL PROBABILITY DISTRIBUTION

  • Extends binomial distribution to k different outcomes.

  • Formula: P = n! / (x1! * x2! * ... * xk!) * (p1^x1 * p2^x2 * ... * pk^xk)

ACTIVITIES AND EXAMPLES

  • Provided activities illustrate application of distributions in real-life scenarios (e.g., probability distribution tables, evaluating success in experiments).