AIM:  Define, calculate and interpret the variance of a random variable.

The expectation of a random variable X is called the mean and typically denoted by Greek mu (µ). If X is a continuous random variable having a density function f(x), then the variance is given by:

But if X is a discrete random variable, the variance is given by:

And the standard deviation, which is simply the square root of the variance, is given by:

Important variance formula

Variance is also conveniently expressed as the difference between the expected value of X^2 and the square of the expected value of X:

• Tip:  memorize this variance formula: it comes in very hand in many situations! For example, if the probability of loan default (PD) is a Bernoulli trial , what is the variance of PD? We can solve with E[PD^2] – (E[PD])^2.

For Example

For example, what is the variance of a single six-sided die? First, we need to solve for the expected value of X-squared, E[X²]. This is given by:

Then, we need to square the expected value of X, [E(X)]². The expected value of a single six-sided die is 3.5 (the average outcome). So, the variance of a single six-sided die is given by:

Here is the derivation of the variance of a single six-sided die (which has a uniform distribution):

• What is the variance of the total of two six-sided die cast together? It is simply the Variance (X) plus the Variance (Y) or about 5.83. The reason we can simply add them together is that they are independent random variables.

Properties of variance

 

• Tip:  Memorize these theorems. These are independent variables so they have zero correlation. The correlation/covariance term drops out, that’s why the variances are additive. Notice that the variance equation is the same for both (X+Y) and (X-Y).

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