Formula to Find E(aX + b) : Proof and Examples

This section introduces the rule for calculating the expected value of a linear function, E(aX + b), stating that if 'a' and 'b' are constants, then E(aX + b) = aE(X) + b. It highlights that this rule simplifies calculations when E(X) is already known.

The video provides a detailed proof of the E(aX + b) formula for a continuous random variable X. It uses integration, the definition of expected value, and properties of integrals to derive E(aX + b) = aE(X) + b.

This part demonstrates how to apply the rule for E(aX + b) to a discrete random variable. It solves an example involving E(2X² + 1), showing how to break down the expression and calculate E(X²) separately using summation.

The video presents an example for a continuous random variable with a given probability density function (PDF). It shows how to calculate E(X^r) using integration, simplifying the expression to a general formula.

This section uses the general formula derived in the previous part to find the expected value of a more complex function, E((2X + 1)²). It expands the expression, separates the terms, and applies the calculated values for E(X²) and E(X).