Sanderson M. Smith

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

mX+Y = mX + mY

mX-Y = mX - mY

If X and Y are independent random variables, then

s2X+Y = s2X + s2Y

s2X-Y = s2X + s2Y (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 X-Y 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.

 X Y X+Y X-Y 2 1 3 1 4 3 5 -1 6 5 7 -3 8 5 3 7 1 9 -1 7 5 9 3 11 1 9 7 11 5 13 3 m 5 3 8 2 s 2.23607 1.63299 2.76887 2.76887 s2 5 2.66667 7.66667 7.66667

 Distribution of X+Y * * * * * * * * * * * * -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Distribution of X-Y * * * * * * * * * * * * -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13

Repeating what is stated above, you should note from this example that the parameters m (mean) and s2 (variance) "behave" quite nicely, but the parameter s (standard deviation) does not. From algebra and geometry, we know that if a2 + b2 = c2, it does not follow that a + b = c.