Sanderson M. Smith
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PROPERTIES OF VARIANCES & MEANS FOR TWO INDEPENDENT RANDOM VARIABLES
Here is an illustration of these important statistical formulas:
If X and Y are any two random variables, then
m_{X+Y} = m_{X} + m_{Y}
m_{XY} = m_{X}  m_{Y}
If X and Y are independent random variables, then
s^{2}_{X+Y} = s^{2}_{X} + s^{2}_{Y}
s^{2}_{XY} = s^{2}_{X} + s^{2}_{Y} (Note: Variances are added, not subtracted.)
Note: These "nice" rules do not hold for standard deviations, as the example below will illustrate.
Now, let X be a member of the set {2,4,6,8} and Y be a member of the set {1,3,5}. Then X+Y is a member of the set {2+1, 2+3, 2+5, 4+1, 4+3, 4+5, 6+1, 6+3, 6+5, 8+1, 8+3, 8+5}. This set contains only six numbers, but one must consider frequencies when calculating means and variances. The set containing values of XY is constructed in a similar manner.
Don't make the careless mistake of thinking the set X+Y consists of {1,2,3,4,5,6,8}. Sometimes you do combine two sets to form a single set, but that's not what is being done here.
You can verify the computations below using your calculator.


Repeating what is stated above, you should note from this example that the parameters m (mean) and s^{2} (variance) "behave" quite nicely, but the parameter s (standard deviation) does not. From algebra and geometry, we know that if a^{2} + b^{2} = c^{2}, it does not follow that a + b = c.
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