What is Expected Value?

Definition:

What is Expected Value?

The Expected Value is the weighted average of the possible outcomes of a random variable, where the weights are the probabilities that the outcomes will occur. The concept of expected value may be demonstrated using probabilities associated with a coin toss. On the flip of one coin, the occurrence of the event "heads" may be used to assign the value of one to a random variable. Alternatively, event "tails" means the random variable equals zero. Statistically, we would formally write the following: if heads, then X = 1; if tails, then X = 0

Example of Expected Value:
The mathematical representation for the expected value of random variable X is:

$ E(X) = \sum P\left ( x_{1} \right )x_{1}+ P\left ( x_{2} \right )x_{2}+…P\left ( x_{n} \right )x_{n} $

where,
E: expectations operator, i.e. used to indicate the computation of a probability-weighted average.
x1: first observed value for random variable x;
x2: second observation, and so on through the nth observation

To calculate the EV for a single discrete random variable, you must multiply the variable's value by the probability of that value occurring. Take, for example, a normal six-sided die. Once you roll the die, it has an equal one-sixth chance of landing on one, two, three, four, five, or six.

Given this information, the calculation is straightforward:

$ \left ( \tfrac{1}{6} x1 \right )+ \left ( \tfrac{1}{6} x2 \right )+ \left ( \tfrac{1}{6} x3 \right )+ \left ( \tfrac{1}{6} x4 \right )+ \left ( \tfrac{1}{6} x5 \right ) = 3.5 $

Why is calculating the expected value important?

In many applications, the expected value plays a crucial role. We want a formula for the observed data that produces a decent estimate of the parameter describing the effect of several explanatory variables on a dependent variable when we undertake regression analysis. In the modern portfolio theory, the risk-neutral agents decide based on the expected values of unknown quantities. On the other hand, it entails maximising the expected value of some objective function for risk-averse actors.