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# Sums and Differences of Independent Random Variables

## Horizontal and vertical translations of the histogram

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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.

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

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:

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

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 Y\begin{align*}Y\end{align*}.
μY=3\begin{align*}\mu_Y=3\end{align*}
σY=1\begin{align*}\sigma_Y=1\end{align*}

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

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

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

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