Expected Value
For taking decisions under conditions of uncertainty, the concept of expected value of a random variable is used. The expected value is the mean of a probability distribution. The mean is computed as the weighted average of the value that the random variable can assume. The probabilities assigned are used as weights. Thus, it is computed by summing up the random variables multiplied by their respective probabilities of occurrence.
E[X] = SX P(X)
Example
A person expects a gain of Rs.80, Rs.120, Rs.160 and Rs.20 by investing in a share. The probability distribution of the gains is as follows.
|
Gain (Rs.)
|
Probability
|
|
80
120
160
20
|
0.2
0.4
0.3
0.1
|
The expected gain from the share is,
(80 x 0.2) + (120 x 0.4) + (160 x 0.3) + (20 x 0.1)
= Rs.(16 + 48 + 48 + 2) = Rs.114
This expected value can be used to compare different investment opportunities. Suppose the investor could invest the amount in another security for which the probability distribution of gains is as follows:
|
Gain (Rs.)
|
Probability
|
|
150
80
20
|
0.1
0.8
0.1
|
The expected gain from the second security is,
(150 x 0.1) + (80 x 0.8) + (20 x 0.1)
= Rs.(15 + 64 + 2) = Rs.81
Since the expected gain from the second security is only Rs.81 as compared to Rs.114 from the first, the investor would do well to invest in the first security.
REMARKS
The points to be noted are:
-
The expected value calculation does not predict the value.
It does not mean that investment in the first security will always lead to a gain of Rs.114 and investment in the second security will always lead to a gain of Rs.81.
-
Comparing the two expected values and taking a decision based on them only helps in ascertaining which of the alternatives is more likely to lead to higher profits.
Since the expected value of gain from the first security is higher than the expected value of gain from the second, one may conclude that the chance of higher gain is more likely from investing in the first rather than the second.