Any random variable has multiple outcomes, which represents by a certain value of possibility. Expect value gives us expected value on a particular event over the mean value of a large set of independent outcomes of the random variable. As we understand that the measures of a particular event over a specific instance tell us about the probability, whereas expected values tell us about the average outcome of a random variable.
What is Expected Value?
We know that the probability distribution is a set of random variables, and the mean of random variables is called the expected value. The expected value naturally refers to what anyone desired or expect if an experiment is repeated infinite times. Mathematically, it is the average of all the possible outcomes of a random experiment; it doesn’t mean the event with the highest probability is the expected value.
How to calculate the expected value: Expected value of a random variable can be calculated in the following ways with certain considerations:
Let x is a discrete random variable and p(x) is the probability of random variable, mean is denoted by μ.
- Fundamental formula for expected value
It is the sum of the probability of the random variable and the number of times the experiment happens:
Expected value, (μ)=xP(x)
Example: Considering a dice is rolled, so the probability of which number comes on its face?
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| P(x) | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |
Expected value,
() = 1*(1/6) + 2*(1/6) + 3*(1/6) + 4*(1/6) + 5*(1/6) + 6*(1/6) = 3.5
- Calculation of expected value for binomial random variables
It is the multiplication of the number of trials and probability of success event.
Example: A coin is tossed 5 times and the probability of getting a tail in each trial is 0.5.
So, Number of trials (X) = 5, and Probability of success event = 0.5.
Expected value = X*P(X) = 5 * 0.5 = 2.5
- Calculation of expected value for multiple events:
Consider a situation; you are an investor for 2 different companies (event). Both companies handle two projects simultaneously with different probabilities so that achieving success by different success values.
Company 1 has two different strategies of probability 0.3 and 0.7 to achieve project value worth $3000 and $4000 respectively.
Company 2 has two different strategies of probability 0.25 and 0.75 to achieve project value worth $3000 and $4000 respectively.
For such scenario, the expected values are as follows:
Expected value for company 1 = 0.3*$3000 + 0.7*$4000 = $3700;
Expected value for company 2 = 0.25*$3000 + 0.75*$4000 = $3750;
Since, EV (Company 2) > EV (Company 1)
$3750 > $3700
With the expected value, we can say that Company B has a better strategy than company A.
- Calculation of expected value for a continuous random variable: If possible values consist of either a particular interval on the number line or a combination of disjoint intervals. The expected value of the continuous random variable is the average of a random variable. Where probability density function is defines the area under the function or curve.
Mathematically: A continuous function f(a) is varies from minus infinity to plus infinity and a is the random variable.
Expected value, Ea=-∞+∞a.fa.da
- Calculation of expected value for an Arbitrary Function: Expected value for an arbitrary function is quite similar to the continuous random variable with certain consideration that is the event is represented by a function f(a) of a random variable g(a) instead of only a.
Simply: a random variable a is replaced by random variable g(a).
Expected value, E[g(a)] = -∞+∞a.fa.da
The expected value has an essential role, especially in statistics and decision theory. An unbiased estimated value of the calculation is exactly equal to the desired parameter value. On the other hand, in decision theory, it is the exact choice of uncertainty. Simply, we can say the expected value of an independent random variable, is the precise value of the arithmetic mean of the result. Also, we have calculated the expected value in the different statistics scenario probability distribution function.
Daily life problem
The price of shooting a target is $5, there are 50 different targets are in the wall. If you successfully hit the target you will get the price of $100 for a particular target out of 50. If you miss the target you will lose the $5 for a chance. What is the expected value for such a scenario?
| Outcome | Price money | Probability |
| Hit the target | +$100 | 1/50 |
| Miss the target | -$5 | 49/50 |
Expected Value = Price money * Probability (Hit the target)
+ Price money * Probability (Miss the target);
Expected Value = $100(1/50) – $5(49/50) = $2 – $4.9 = -$2.9