Discrete Random Variables

Overview of Discrete Random Variables

Goals of the Video:
- Understand what a discrete random variable is.
- Learn the probability distribution of a random variable.
- Know about the mean and variance of a discrete random variable.

Definition: A random variable assigns numerical values to each outcome of a random experiment.
- Example: In flipping three pennies, let X be the number of heads.
Possible outcomes for flipping three coins:
- 0 Heads: 1 way (TTT)
- 1 Head: 3 ways (HTT, THT, TTH)
- 2 Heads: 3 ways (HHT, HTH, THH)
- 3 Heads: 1 way (HHH)

Definition: A listing or function indicating the probabilities associated with each possible value that the random variable can take.
- Mathematical representation:
  - f = Probability P(X = x)
For the three pennies flip, the probability distribution is:
- P(X=0) = 1/8
- P(X=1) = 3/8
- P(X=2) = 3/8
- P(X=3) = 1/8
Properties of Probability Distribution:
- Each probability must be non-negative: P(X=x) ≥ 0 for all x.
- The total probability must equal 1: Σ P(X=x) = 1.

Purpose: Measure variability in the probability distribution.
Calculation of Variance:
- Variance (Var(X) or σ²) is computed as:
- Formula: Var(X) = Σ (x² * P(X=x)) - μ²
- Alternatively, calculate E(X²) and then determine variance using: Var(X) = E(X²) - μ².
Example of Variance Calculation:
- Using tables for two games to illustrate.
- Game 1:
  - Possible values: -2 (1/4), 0 (1/2), 2 (1/4)
  - Calculation yields E(X²) = 2.
- Game 2:
  - Possible values: -4 (1/4), 0 (1/2), 4 (1/4)
  - Calculation yields E(X²) = 8.
Standard Deviation:
- Standard deviation (σ) is the square root of the variance.
- Game 1: σ = √2; Game 2: σ = 2√2.
- Game 2's standard deviation is higher, indicating greater risk compared to Game 1.

Understanding discrete random variables helps in grasping the concepts of probability distribution, mean, and variance, which are essential for analyzing random processes and making informed decisions.