EXPECTED VALUE (THE MEAN) AND VARIANCE OF A DISCRETE PROBABILITY
DISTRIBUTION

THE EXPECTED VALUE

Assume the probability distribution of heads for the experiment consisting of flipping an unbiased coin three times: 
Pr (X = 0) = 1/8
Pr (X = 1) = 3/8
Pr (X = 2) = 3/8
Pr (X = 3) = 1/8.
If this experiment is performed a large number of times, the value of X will bounce around randomly according the probability distribution above. Now assume that we wish to find a central value (a specific number) around which X randomly fluctuates. By looking at the distribution above, a good guess at what this number is would be 1.5. Notice that the number "2" is as much above 1.5 as the number "1" is below it and they have the same probability "weights". The same argument can be made for "0" and "3". That is, the number "0" is as much below 1.5 as the number "3" is above it and each number has the same probability "weight". We say that there is symmetry around 1.5 since, in a real sense, it is the number in the center and the different values of X fluctuate around it. There is a simple formula to calculate this central number. It is:

E(x) = µ = x* P(x) = x * P(x) + x * P(x) + ........

Applying this formula to the distribution above, we obtain

E(X) = µ = 0*1/8 + 1*3/8 + 2*3/8 + 3*3/8 = 1.5.

THE VARIANCE OF A PROBABILITY DISTRIBUTION

Again assume the probability distribution of heads for the experiment consisting of flipping an unbiased coin three times:

Pr (X = 0) = 1/8
Pr (X = 1) = 3/8
Pr (X = 2) = 3/8
Pr (X = 3) = 1/8.

The variance of a probability distribution is a measure not of its central value but of the dispersion around the expected value. It is a measure of how "spread out" the X values are around the center or expected value. The formula for the variance is:

VAR(x) = = (x - µ) P(x) = (x - µ) *P(x) + (x - µ) * P(x) + ........

An alternative formula for the Variance is:

VAR(x) = = E(x) - [µ]

This is usually easier computationally.

TABLE FORMAT FOR CALCULATION OF EXPECTED VALUE AND VARIANCE: EXAMPLE: PROBABILITY DISTRIBUTION OF NUMBER OF HEADS IN THREE COIN FLIP EXPERIMENT 
1	2	3	4	5		6		7	
X	P(X)	X*P(X) X - µx (X -µx)2       (X-µx)2*P(X)     X2P(X)
0	1/8	0	-3/2	9/4		9/32		0
1	3/8	3/8	-1/2	1/4		3/32		3/8
2	3/8	6/8	+1/2	1/4		3/32		12/8
3	1/8	3/8	+3/2	9/4		9/32		9/8
		___				____		___
SUMS:		µx = 12/8			sx2 =24/32=.75  24/8= 3
NOTES FOR TABLE:
1. COLUMN 7 IS OBTAINED BY MULTIPLYING COLUMNS 1 AND 3.
2. NOTE THAT THE APPLICATION OF THE ALTERNATIVE FORMULA FOR THE COMPUTUTATION OF THE VARIANCE YIELDS 3 - 2.25 = .75.

RULES FOR EXPECTED VALUE AND VARIANCE:
1. IF X IS A RANDOM VARIABLE (RV), THEN a*X WHERE a IS A CONSTANT HAS AN EXPECTED VALUE EQUAL TO a TIMES THE EXPECTED VALUE OF X AND A VARIANCE EQUAL TO a2 TIMES THE VARIANCE OF X. 2. IF X AND Y ARE RANDOM VARIABLES, THEN

E(X + Y) = E(X) + E(Y), E(X - Y) = E(X) - E(Y)

AND, IF X AND Y ARE STATISTICALLY INDEPENDENT, THEN

VAR(X + Y) = VAR (X + Y) AND VAR(X - Y) = VAR (X + Y).

3. THE EXPECTED VALUE OF A CONSTANT IS EQUAL TO THE CONSTANT AND THE VARIANCE OF A CONSTANT IS EQUAL TO ZERO.

4. IF X AND Y ARE RV'S AND A NEW RV Z = aX + bY IS FORMED WHERE a AND b ARE CONSTANTS, THEN:
A. E(Z) = aE(X) + bE(Y).
B. IF Z = aX - bY, E(Z) = aE(X) - bE(Y).
C. IF X AND Y ARE STATISTICALLY INDEPENDENT, AND Z = aX + bY, THEN

VAR(Z) = a * VAR(X) + b * VAR(Y).

D. IF Z = aX - bY, X AND Y STATISTICALLY INDEPENDENT, THEN

VAR(Z) = a * VAR(X) + b * VAR(Y).