Business Analytics > Probability, Risk & Sampling > Expected Value of a Discrete Random Variable
Expected Value of a Discrete Random Variable
The expected value of a random variable corresponds to the notion of the mean, or average, for a sample. For a discrete random variable X, the expected value, denoted E[X], is the weighted average of all possible outcomes, where the weights are the probabilities:
A business launches a promotion where profit of 100,000 with probability 0.6 and loss of 50,000 with probability 0.4. So the expected value is
EV=(100,000×0.6)+(−50,000×0.4)
EV=60,000−20,000=40,000
Expected value = 40,000
Example: The previous statistics says that Bangladesh cricket team got wickets in the world cup semi-final zero, one, two, three, four, five, six, seven .. ten wickets with the following probabilities.
|
Number of wickets |
Probabilities |
|
0 |
0.01 |
|
1 |
0.05 |
|
2 |
0.09 |
|
3 |
0.12 |
|
4 |
0.17 |
|
5 |
0.16 |
|
6 |
0.14 |
|
8 |
0.11 |
|
8 |
0.09 |
|
9 |
0.04 |
|
10 |
0.02 |
Find the long-term average or expected value, μ, of the number of wickets get per play the cricket team.
Solution:
|
Number of wickets (x) |
Probabilities p(x) |
|
|
0 |
0.01 |
0 |
|
1 |
0.05 |
0.05 |
|
2 |
0.09 |
0.18 |
|
3 |
0.12 |
0.36 |
|
4 |
0.17 |
0.68 |
|
5 |
0.16 |
0.8 |
|
6 |
0.14 |
0.84 |
|
8 |
0.11 |
0.88 |
|
8 |
0.09 |
0.72 |
|
9 |
0.04 |
0.36 |
|
10 |
0.02 |
0.2 |
|
Total = |
5.07 |
|
Expected value = 5.07 i.e. average number number of wickets per play is 5.07.
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