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.

Discrete Random Variables

• 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)

Probability Distribution

• 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.

Mean of a Discrete Random Variable

• Calculation of Mean:

Variance of a Discrete Random Variable

• 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.

Conclusion

• 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.