How is the expected value of a random variable calculated?

The expected value of a random variable, often denoted as E(X), represents the long-term average outcome of a random variable based on its probability distribution. It is computed by multiplying each possible value of the random variable by the probability of that value occurring and then summing these products.

In this context, the formula E(X) = x1p1 + x2p2 + ... + xnpn precisely captures this process. Here, x1, x2, ..., xn are the possible values of the random variable, while p1, p2, ..., pn are their corresponding probabilities. This formula accounts for each outcome and how likely it is to occur, thus giving a weighted average of all possible values.

The other options do not properly describe how to compute the expected value. For instance, simply summing probabilities or outcomes without considering their corresponding weights will not yield the correct expected value calculation, as it neglects the essential aspect of the probability linked to each outcome.