Rescaling Expectation, Variance and Std

Rescaling of Expectation, Variance, and Standard Deviation

When a random variable $X$ is rescaled by a constant $a$, where $Y=aX+b$, the expectation, variance, and standard deviation are affected as follows:

• The expectation becomes $E(Y)=aE(X)+b$
• The variance becomes $\text{Var}(Y)=a^2\text{Var}(X)$.
• The standard deviation, being the square root of the variance, is scaled by the absolute value of $a:\text{SD}(Y)=|a|\text{SD}(X)$.