Normal distribution a Var(x) to solve problems. Use E(ax+b) = aE(x) + b. Var(ax+b) = a2 Var(x).

The lesson plan outlines the following key properties of expectation and variance for random variables:

Step 1: Property of Expectation for a linear transformation of a single random variable. For a random variable $X$ and constants $a$ and $b$, the expectation of $aX+b$ is given by: $E(aX+b) = aE(X) + b$

Step 2: Property of Expectation for the sum of two random variables. For two random variables $X$ and $Y$, the expectation of their sum is given by: $E(X+Y) = E(X) + E(Y)$ This property holds true regardless of whether $X$ and $Y$ are independent.

Step 3: Property of Variance for a linear transformation of a single random variable. For a random variable $X$ and constants $a$ and $b$, the variance of $aX+b$ is given by: $Var(aX+b) = a^{2}Var(X)$

Step 4: Property of Variance for a linear combination of two independent random variables. For two independent random variables $X$ and $Y$, and constants $a$ and $b$, the variance of their linear combination $aX+bY$ is given by: $Var(aX+bY) = a^{2}Var(X) + b^{2}Var(Y)$ This formula specifically applies when $X$ and $Y$ are independent.