How to Calculate Expected Value of X^2

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For a random variable, denoted as X, you can use the following formula to calculate the expected value of X²:

E(X²) = Σx² * 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 X²

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 X², we can use the following formula:

E(X²) = Σx² * p(x)

E(X²) = (0)²*.06 + (1)²*.15 + (2)²*.17 + (3)²*.24 + (4)²*.23 + (5)²*.09 + (6)²*.06

E(X²) = 0 + .15 + .68 + 2.16 + 3.68 + 2.25+ 2.16

E(X²) = 11.08

The expected value of X² 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 X²:

E(X²) = ∫ x²f(x)dx

where:

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

When calculating the expected value of X² 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