Expected values E(X²)

Understanding the Expectation of X² (E(X²))

In the world of statistics, the Expectation of X² (E(X²)) can be a potentially confusing concept. Essentially, it represents the expected value of the square of a random variable X.
The concept is usually introduced to tackle problems that involve the variance and standard deviation of a probability distribution.
E(X²) is not the same as the square of the expectation of X [E(X)]². Thus, E(X²) ≠ [E(X)]². This is a common point of confusion, so always remember that they are not interchangeable.

Calculation of E(X²)

To calculate E(X²), square the values of the random variable outcomes, then multiply each by its respective probability. The sum of these results is E(X²).
For example, if X takes the values of 1, 2, 3 with probabilities of 0.25, 0.50, and 0.25 respectively, then E(X²) = 0.25(1²) + 0.50(2²) + 0.25*(3²).

Applications of E(X²)

E(X²) plays a crucial role when calculating the variance of a random variable X. The variance Var(X) = E(X²) - [E(X)]².
Understanding E(X²) is critical for modelling, interpreting, predicting, and analyzing the behavior of random variables in a wide array of real-life situations, such as risk analyses, scientific research, investment returns, and many others.

Key Points to Remember

Do not confuse E(X²) with the square of the expectation of X, [E(X)]².
E(X²) is not necessarily a measure of ‘average’ in the conventional sense, but rather the expected value of the squared outcomes.
A solid understanding of the concept of E(X²) is necessary for a deeper appreciation and application of Advanced Statistics.