The formula to determine the expected value E(X)E(X)E(X) of a discrete random variable XXX is:

E(X)=∑i=1nxi⋅P(xi)E(X)=\sum_{i=1}^nx_i\cdot P(x_i)E(X)=i=1∑n​xi​⋅P(xi​)

where:

• xix_ixi​ are the possible values of the random variable,
• P(xi)P(x_i)P(xi​) is the probability of each value xix_ixi​,
• nnn is the number of possible outcomes.

In words, the expected value is calculated by multiplying each possible outcome by its probability and then summing all those products. This gives the weighted average or mean of the random variable's outcomes over the long run

. For example, if you have a random variable with values x1,x2,...,xnx_1,x_2,...,x_nx1​,x2​,...,xn​ and corresponding probabilities p1,p2,...,pnp_1,p_2,...,p_np1​,p2​,...,pn​, then:

E(X)=x1p1+x2p2+⋯+xnpnE(X)=x_1p_1+x_2p_2+\cdots +x_np_nE(X)=x1​p1​+x2​p2​+⋯+xn​pn​

This formula applies broadly in probability, statistics, and applications such as investment decision-making, where expected value helps evaluate the average return considering all possible outcomes and their likelihoods