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Expected Value and Variance of Sums

When you’ve already found the expected value and variance of a data set, it’s good to know that you won’t have to calculate everything all over again if you make small changes to the data. You can actually use the expected value and variance you’ve already found by inserting them into certain formulas, which I will show you here.

Theory

Sum of Variables

A sum of random variables might look like this:

Z = a X + Y + b

where $X$ and $Y$ are random variables that are independent of each other, and $a$ and $b$ are constants.

When you have values for $E (X)$ , $E (Y)$ , $Var (X)$ and $Var (Y)$ , you can use them directly if you have a sum of variables. An otherwise gnarly calculation becomes a lot easier with these formulas:

Formula

Useful Formulas with the Sum of Variables

\begin{array}{l} E (Z) & = E (a X + Y + b) \\ = E (a X) + E (Y + b) \\ = a E (X) + E (Y) + b \\ Var (Z) & = Var (a X + Y + b) \\ = Var (a X) + Var (Y + b) \\ = a^{2} Var (X) + Var (Y) \end{array}

Note! The constant term disappears when you find the variance.

Example 1

Given $Z = 3 X + Y + 6$ and $E (X) = 7$ , $E (Y) = 16$ , $Var (X) = 2$ and $Var (Y) = 3$ , find the expected value $E (Z)$ and the variance $Var (Z)$ .

You find the expected value by inserting what we know directly into the formulas:

\begin{array}{l} E (Z) & = E (3 X + Y + 6) \\ = E (3 X) + E (Y + 6) \\ = 3 E (X) + E (Y) + 6 \\ = 3 \cdot 7 + 16 + 6 \\ = 43 \end{array}

You find the variance by doing the same thing:

\begin{array}{l} Var (Z) & = Var (3 X + Y + 6) \\ = Var (3 X) + Var (Y + 6) \\ = 3^{2} Var (X) + Var (Y) \\ = 9 \cdot 2 + 3 \\ = 21 \end{array}

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