PROPERTIES OF VARIANCES & MEANS FOR TWO INDEPENDENT RANDOM VARIABLES

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

m_X+Y = m_X + m_Y

m_X-Y = m_X - m_Y

If X and Y are independent random variables, then

s²_X+Y = s²_X + s²_Y

s²_X-Y = s²_X + s²_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 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.

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