Lecture 2

Title: STA013 Lecture 5: Discrete Probability DistributionsUniversity: UC Davis

Introduction

Random Variable: A variable determined by a random process.
Key Characteristics: Defined by probability distribution; focus on binomial distribution.

Contents Overview

Random Variables

Definition: Variable X as a random variable if its value correlates to a random experiment outcome.
Types:
- Discrete: Finite or countably infinite values.
- Continuous: Any value within a range.
Importance: Different techniques for describing distributions for discrete vs. continuous variables.

Probability Distribution

Definition: Relative frequency distribution across the population of measurements, including possible values of X and corresponding probabilities P(X=x).
Requirements:
- Probability values: 0 ≤ p(x) ≤ 1.
- Total probability must equal 1.

Example: Two Fair Coins - Probability distribution for number of heads when tossing two coins. Confirming probabilities sum to 1.

Mean and Standard Deviation

Mean (E(X)): Average value calculated as μ = Σx p(x).
Variance (σ²): Measures variability; standard deviation (σ) is the square root of variance.
Example: Daily laptop demand represented in probability distribution with results μ = 1.90 and P(X ≥ 5) = 0.05.

Binomial Distribution

Characteristics: n identical trials, outcomes as success (S) or failure (F), constant success probability p, independent trials.
Random Variable: X represents the successes in n trials.
Example: Study on adults' support for term limits; fit characteristics for binomial experiment.
Formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k).
Cumulative Probability: Defined as F(x) = P(X ≤ x).
Application: Evaluate scenarios with binomial calculations, e.g., effectiveness of vitamin C against colds.