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Sums and Differences of Independent Random Variables

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Transforming Random Variables

When you add or subtract a constant from the outcomes of a random variable, the mean will increase or decrease by that amount, and the standard deviation will stay the same.

Example:  This distribution is called X.

\begin{align*}\mu_X &=3 \\ \sigma_X &=1\end{align*}μXσX=3=1

If we add 2 to the distribution, it becomes X + 2.  Now the distribution has a mean of 5, but the standard deviation is still 1:

\begin{align*}\mu_{X+2} &=5 \\ \sigma_{X+2} &=1\end{align*}μX+2σX+2=5=1

When you multiply or divide the outcomes of a random variable by a constant, the mean and the standard deviation will also be multiplied or divided by that constant.

Example: This distribution is called \begin{align*}Y\end{align*}Y.
\begin{align*}\mu_Y=3\end{align*}μY=3 
\begin{align*}\sigma_Y=1\end{align*}σY=1 

Let’s see what happens if we multiply each outcome by 3, and find \begin{align*}\mu_{3Y}\end{align*}μ3Y and \begin{align*}\sigma_{3Y}\end{align*}σ3Y :

 \begin{align*}\mu_{3Y} = 3 \times 3=9\end{align*}μ3Y=3×3=9 

\begin{align*}\sigma_{3Y}=1 \times 3=3\end{align*}σ3Y=1×3=3 

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