STAT 330 - CHAPTER 4 | GUARANTEED SUCCESS STARTS
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Q: What is the expected value (mean) of a discrete random variable? Answer: E(X) = Σ x f(x).
Q: What is the expected value of a continuous random variable? Answer: E(X) = ∫ x f(x) dx.
Q: What is the variance of X? Answer: Var(X) = E[(X - μ)²] = E(X²) - μ².
Q: What is the standard deviation of X? Answer: σ = √Var(X).
Q: What is the kth moment about the origin? Answer: μ'k = E(X^k).
Q: What is the kth central moment? Answer: μk = E[(X - μ)^k].
Q: What is skewness? Answer: A measure of asymmetry: γ1 = μ3 / σ³.
Q: What does positive skewness mean? Answer: Distribution tail is longer on the right.
Q: What does negative skewness mean? Answer: Distribution tail is longer on the left.
Q: What is kurtosis? Answer: A measure of peakedness: γ2 = μ4 / σ⁴.
Q: What is excess kurtosis? Answer: γ2 - 3 (compared to normal distribution).
Q: What is the expected value of a linear function aX+b? Answer: E(aX+b) = aE(X)+b.
Q: What is the variance of a linear function aX+b? Answer: Var(aX+b) = a² Var(X).
Q: What is the expected value of aX+bY? Answer: E(aX+bY) = aE(X)+bE(Y).
Q: What is covariance? Answer: Cov(X,Y) = E[(X - μX)(Y - μY)].
Q: What is correlation? Answer: ρ = Cov(X,Y)/(σXσY), between -1 and 1.
Q: What does ρ = 1 mean? Answer: Perfect positive linear relationship.
Q: What does ρ = -1 mean? Answer: Perfect negative linear relationship.
Q: What does ρ = 0 mean? Answer: No linear correlation (but not necessarily independent).
Q: What is the variance of the sum of two independent RVs? Answer: Var(X+Y) = Var(X)+Var(Y).
Q: What is the expected value of the sum of two RVs? Answer: E(X+Y) = E(X)+E(Y).
Q: Example: Roll a fair die (X = outcome). What is E(X)? Answer: E(X) = 3.5.
Q: Example: Roll a fair die (X = outcome). What is Var(X)? Answer: Var(X) = 35/12 ≈ 2.92.
T: Expected Value Answer: The long-run average value of a random variable.
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HERE! LEARN, PRACTICE & EXCEL!
Q: What is the expected value (mean) of a discrete random variable? Answer: E(X) = Σ x f(x).
Q: What is the expected value of a continuous random variable? Answer: E(X) = ∫ x f(x) dx.
Q: What is the variance of X? Answer: Var(X) = E[(X - μ)²] = E(X²) - μ².
Q: What is the standard deviation of X? Answer: σ = √Var(X).
Q: What is the kth moment about the origin? Answer: μ'k = E(X^k).
Q: What is the kth central moment? Answer: μk = E[(X - μ)^k].
Q: What is skewness? Answer: A measure of asymmetry: γ1 = μ3 / σ³.
Q: What does positive skewness mean? Answer: Distribution tail is longer on the right.
Q: What does negative skewness mean? Answer: Distribution tail is longer on the left.
Q: What is kurtosis? Answer: A measure of peakedness: γ2 = μ4 / σ⁴.
Q: What is excess kurtosis? Answer: γ2 - 3 (compared to normal distribution).
Q: What is the expected value of a linear function aX+b? Answer: E(aX+b) = aE(X)+b.
Q: What is the variance of a linear function aX+b? Answer: Var(aX+b) = a² Var(X).
Q: What is the expected value of aX+bY? Answer: E(aX+bY) = aE(X)+bE(Y).
Q: What is covariance? Answer: Cov(X,Y) = E[(X - μX)(Y - μY)].
Q: What is correlation? Answer: ρ = Cov(X,Y)/(σXσY), between -1 and 1.
Q: What does ρ = 1 mean? Answer: Perfect positive linear relationship.
Q: What does ρ = -1 mean? Answer: Perfect negative linear relationship.
Q: What does ρ = 0 mean? Answer: No linear correlation (but not necessarily independent).
Q: What is the variance of the sum of two independent RVs? Answer: Var(X+Y) = Var(X)+Var(Y).
Q: What is the expected value of the sum of two RVs? Answer: E(X+Y) = E(X)+E(Y).
Q: Example: Roll a fair die (X = outcome). What is E(X)? Answer: E(X) = 3.5.
Q: Example: Roll a fair die (X = outcome). What is Var(X)? Answer: Var(X) = 35/12 ≈ 2.92.
T: Expected Value Answer: The long-run average value of a random variable.
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