The expected value of a linear transformation of a random variable $X$ is given by:
$$E(aX + b) = aE(X) + b$$
This formula is incredibly useful as it allows us to calculate the expected value of transformed data without having to recalculate the entire distribution.
Suppose we have three random variables: \[\begin{aligned} &X_1: \text{Number of cars sold}, \quad E(X_1)=10 \\ &X_2: \text{Number of motorcycles sold}, \quad E(X_2)=5 \\ &X_3: \text{Number of bicycles sold}, \quad E(X_3)=20 \end{aligned}\] If the profit for each is $1000, $500, and $200 respectively, then the expected total profit is $$E\left(1000X_1+500X_2+200X_3\right)=1000E\left(X_1\right)+500E\left(X_2\right)+200E\left(X_3\right)=1000(10)+500(5)+200(20)=16{,}500\text{ dollars}.$$
The variance of a linear transformation of a random variable $X$ is given by:
$$VAR(aX + b) = a^2 VAR(X)$$
Notice that the constant $b$ doesn't affect the variance, as it doesn't contribute to the spread of the data.
The standard deviation of a linear transformation is the absolute value of $a$ times the standard deviation of $X$: $SD(aX + b) = |a| \cdot SD(X)$.
Continuing with our height example, if $VAR(X) = 100$ cm², then:
$VAR(0.3937X + 2) = 0.3937^2 * 100 ≈ 15.5$ inches²
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