For a random variable, denoted as X, you can use the following formula to calculate the expected value of X^{2}:

**E(X ^{2}) = Σx^{2} * p(x)**

where:

**Σ**: A symbol that means “summation”**x**: The value of the random variable**p(x)**:The probability that the random variable takes on a given value

The following example shows how to use this formula in practice.

**Example: Calculating Expected Value of X**^{2}

^{2}

Suppose we have the following probability distribution table that describes the probability that some random variable, X, takes on various values:

To calculate the expected value of X^{2}, we can use the following formula:

E(X^{2}) = Σx^{2} * p(x)

E(X^{2}) = (0)^{2}*.06 + (1)^{2}*.15 + (2)^{2}*.17 + (3)^{2}*.24 + (4)^{2}*.23 + (5)^{2}*.09 + (6)^{2}*.06

E(X^{2}) = 0 + .15 + .68 + 2.16 + 3.68 + 2.25+ 2.16

E(X^{2}) = 11.08

The expected value of X^{2} is **11.08**.

Note that this random variable is a **discrete random variable**, which means it can only take on a finite number of values.

If X is a **continuous random variable**, we must use the following formula to calculate the expected value of X^{2}:

**E(X ^{2}) = ∫ x^{2}f(x)dx**

where:

- ∫ : A symbol that means “integration”
**f(x)**: The continuous pdf for the random variable X

When calculating the expected value of X^{2} for a continuous random variable, we typically use statistical software since this computation can be more difficult to perform by hand.

**Additional Resources**

The following tutorials explain how to perform other common tasks in statistics:

How to Find the Mean of a Probability Distribution

How to Find the Standard Deviation of a Probability Distribution

How to Find the Variance of a Probability Distribution