Introduction to Product Moments (About the Origin and About the Mean)

The video introduces the concept of product moments, extending the idea of moments from one random variable to two. It distinguishes between product moments about the origin and product moments about the mean. The definitions are provided for both discrete (double summation) and continuous (double integral) random variables.

The notation for the r-th and s-th product moment about the origin is introduced as μ'rs, where 'r' relates to random variable X and 's' to random variable Y. This is defined as the expected value of X^r * Y^s. For discrete variables, it involves a double summation, and for continuous variables, a double integral of X^r * Y^s multiplied by the joint probability function. Specific cases where r=1, s=0 and r=0, s=1 simplify to the mean of X (μx) and the mean of Y (μy), respectively.

The video then moves on to defining the r-th and s-th product moments about the mean, denoted as μrs (without the prime). This involves subtracting the respective means from each random variable: E[(X - μx)^r * (Y - μy)^s]. Similar to product moments about the origin, discrete cases use double summation, and continuous cases use double integrals.

A special case of product moment about the mean occurs when r=1 and s=1, which is defined as the covariance of X and Y, denoted as Cov(X, Y). Covariance indicates the direction of the linear relationship between two variables. A positive covariance implies they move in the same direction, while a negative covariance implies they move in opposite directions.

The video demonstrates how to prove the common formula for covariance: Cov(X, Y) = E(XY) - E(X)E(Y). This involves expanding the definition of product moment about the mean when r=1 and s=1, leveraging properties of expectation, and simplifying terms. The proof shows that E(X) and E(Y) are equivalent to μx and μy, and terms cancel out, leading to the desired formula.

The derived covariance formula is compared to the formula for the variance of a single random variable, Var(X) = E(X^2) - (E(X))^2. The structural similarity of the two formulas is highlighted, making the covariance formula easier to remember by relating X^2 to X*X and (E(X))^2 to E(X)*E(X).