This section is about transforming random variables by adding/subtracting or multiplying/dividing by a constant. Make sure you know how to combine random variables to calculate and interpret the mean and standard deviation.

## Linear Transformations of a Random Variable

*Y = a + BX*

Adding/Subtracting by a constant affects** measures of center and location **but does NOT affect variability or the shape of a distribution. Multiplying or dividing by a constant affects **center, location, and variability measures** but won’t change the shape of a distribution.

## How to Find the Expected Value of the Sum/Difference of Two Random Variables

**Sum: **For any two random variables X and Y, **if S = X + Y,** the mean of S is **meanS= meanX + meanY**. Put simply, the mean of the sum of two random variables is equal to the sum of their means.

**Difference: **For any two random variables X and Y, **if D = X - Y, **the mean of D is **meanD= meanX - meanY**. The mean of the difference of two random variables is equal to the difference of their means. *The order of subtraction is important.*

**Independent Random Variables: **If knowing the value of X does not help us predict the value of X, then X and Y are independent random variables.

## Standard Deviation of the Sum/Difference of Two Independent Random Variables

**Sum:** For any two independent random variables X and Y, if S = X + Y, the **variance** of S is SD^2= (X+Y)^2 . To find the **standard deviation, **take the square root of the variance formula: SD = sqrt(SDX^2 + SDY^2).

*Standard deviations do not add; use the formula or your calculator. *

**Difference: **For any two independent random variables X and Y, if D = X - Y, the **variance** of D is D^2= (X-Y)^2=x2+Y2. To find the **standard deviation, **take the square root of the variance formula: D=sqrt(x2+Y2). Notice that you are NOT subtracting the variances (or the standard deviation in the latter formula).

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