Statistics for Maintenance Modelling: Basic Probability Concepts

# Expectation and Variance¤

The expected value and variance are two properties of random variables that help define how the random variable is distributed. The expected value gives an indication of where about the random variable is located, and the variance gives an indication of how the random variable is most likely spread about the expected value. The formula for both the expected value and the variance of a random variable depends on whether the random variable is discrete or continuous.

## Expected Value for Discrete Random Variables

For a discrete random variable with probability mass function p(x), the expected value is given by:

This function says that the expected value of X is a weighted average of all the possible values (n) that X can take on, each value being weighted by the probability mass that X assumes it.

See if you can work out the expected value of X, where X is the outcome when a die is rolled. Type your answer into the textbox and click the "Check Answer" button. Click the "Hint" button for some help in working out the answer.

| E[X] =

Yes, that's right, E[X] = 3.5. This is because for all i, P(xi) = 1/6, so we get:

E[X] = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 7/2

No that's not right. E[X] = 3.5. Since p(1) = p(2) = p(3) = p(4) = p(4) = p(6) = 1/6, we get:

E[X] = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 7/2