*43*

For a random variable, denoted as X, you can use the following formula to calculate the expected value of X^{3}:

**E(X ^{3}) = Σx^{3} * 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**^{3}

^{3}

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^{3}, we can use the following formula:

E(X^{3}) = Σx^{3} * p(x)

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

E(X^{3}) = 0 + .15 + .1.36 + 6.48 + 14.72 + 11.25 + 12.96

E(X^{3}) = 45.596

The expected value of X^{3} is **45.596**.

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^{3}:

**E(X ^{3}) = ∫ x^{3}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^{3} 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