Expected Value of a Random Variable

Expected Value of a random variable

We all understand the concept of average. The average test-score of a class is the sum of each individual student's score divided by the total number of students in the class. The expected value of a random variable is somewhat a similar concept. However, note that the outcomes that a random variable can take on don't happen with the same frequency. An outcome that happens more frequently should get a higher weight when computing the "average" for a random variable. The weight here is the probability with which each outcome occurs.

Take the case of the random variable X which we define as the number of heads that occur in 3 flips of a coin. The expected number of heads seen when the experiment of flipping a coin thrice is repeated many many times is shown below:

$$E[X] = P(X=0) * 0 + P(X=1) * 1 + P(X=2) * 2 + P(X=3) * 3$$ ...