Econ 120A Discrete Probability Distributions

Flashcards covering key vocabulary and formulas related to discrete probability distributions, mean, variance, and expected values.

Population Mean (𝜇)

𝜇 = ෍𝑥𝑝(𝑥), the sum of each value times its probability.

Population Variance (𝜎²)

𝜎² = ෍(𝑥 − 𝜇)²𝑝(𝑥), the sum of the squared difference between each value and the mean, weighted by its probability.

Expectations Operator E(.)

Takes the weighted average of (.), in which the weights are the probabilities.

E(X)

σ 𝑥 𝑝(𝑥) = 𝜇𝑋 The expected value of X, is the weighted average of its possible realizations weighted by their probabilities of occurring; same as the mean.

E[g(X)]

σ 𝑔(𝑥) 𝑝(𝑥) = 𝜇𝑅; The expected value of random variable R (a function of the random variable X) is also its mean.

E(X²)

σ 𝑥²𝑝(𝑥); the expected value of X squared.

E[(X − 𝜇𝑋)²]

σ (𝑥 − 𝜇𝑋)²𝑝(𝑥) = 𝜎𝑋²; the expected value of the squared deviations from the mean of the random variable X, which is its variance.

E(c)

c; the expected value of a constant is the constant itself.

E(X + c)

E(X) + E(c) = E(X) + c; the expected value of a sum of a variable plus a constant is equal to the sum of expected values.

E(cX)

c E(X); the expected value of cX.

Var(X + c)

Var(X); Variance of (X + c)

Var(cX)

c² Var(X); Variance of cX

𝜎𝑋²

E[𝑋 − 𝜇𝑋)²] = 𝐸[𝑋²] − 𝜇𝑋²