Lecture Notes on Expectation, Variance, and Covariance

Recap of Previous Class

• Key concepts covered: Expectation, Variance, and Covariance.

Expectation

• Expectation (E) of a random variable X: (E(X))
  • For discrete random variables (RVs): (\sum_x x \cdot P(x))
  • For continuous RVs: (\int_{-\infty}^{\infty} x \cdot f(x) , dx)
• Example: Given (E(X) = 3/5), determine constants (a) and (b) in the probability density function (f(x) = a + b \cdot x^2) for (0 \leq x \leq 1).
  • Equation 1: (a + b/3 = 1)
  • Equation 2: (2a + b = 12/5)
  • Solution gives: (a = 3/5), (b = 6/5)_

Properties of Expectation

• (E(aX + b) = a \cdot E(X) + b)
• Example calculation involving (E(2 + 4X)^2)
  • Expand and simplify using properties of expectation.

Variance

• Variance (Var): Measures the spread of the distribution
  • (Var(X) = E(X^2) - [E(X)]^2)
• Identity: (Var(aX + b) = a^2 \cdot Var(X))

Covariance

• Covariance (Cov): Measures the joint variability of two random variables X and Y
  • (Cov(X, Y) = E(XY) - E(X)E(Y))
• Special Case: If X and Y are independent, then (Cov(X, Y) = 0).

Calculation of Variance in Specific Scenarios

• Coin Toss Example:
  • Toss a coin 10 times, compute variance of the number of heads.
  • Independent events: (Var(\sum X_i) = \sum Var(X_i))
  • (Var(X_i) = 1/4), hence (Var = 10 \times 1/4 = 5/2)
• Die Roll Example:
  • Roll a die 10 times, compute variance of the sum.
  • (E(X) = 7/2), (Var(X_i)) calculated, (Var = 10 \times 35/12).

Correlation Coefficient

• (\rho = \frac{Cov(X, Y)}{\sqrt{Var(X) \cdot Var(Y)}})

Key Takeaways

• Expectation and Variance: Essential for estimating quantities and understanding distributions.
• Formula Recap:
  • (E(g(X)) = \sum g(x)P(x)) or (\int g(x)f(x) , dx)
  • (Var(aX + b) = a^2 \cdot Var(X))
  • For independent RVs: (Var(\sum X_i) = \sum Var(X_i))

Thank you for your attention. We'll meet again in the next class.