# Conditional variance

In probability theory and statistics, a **conditional variance** is the variance of a conditional probability distribution. That is, it is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the **scedastic function** or **skedastic function**. Conditional variances are important parts of autoregressive conditional heteroskedasticity (ARCH) models.

## Definition

The conditional variance of a random variable *Y* given that the value of a random variable *X* takes the value *x* is

where E is the expectation operator with respect to the conditional distribution of *Y* given that the *X* takes the value *x*. An alternative notation for this is :

The above may be stated in the alternative form that, based on the conditional distribution of *Y* given that the *X* takes the value *x*, the conditional variance is the variance of this probability distribution.

## Components of variance

The law of total variance says

where, for example, is understood to mean that the value *x* at which the conditional variance would be evaluated is allowed to be a random variable, *X*. In this "law", the inner expectation or variance is taken with respect to *Y* conditional on *X*, while the outer expectation or variance is taken with respect to *X*. This expression represents the overall variance of *Y* as the sum of two components, involving a prediction of *Y* based on *X*. Specifically, let the predictor be the least-mean-squares prediction based on *X*, which is the conditional expectation of *Y* given *X*. Then the two components are:

- the average of the variance of
*Y*about the prediction based on*X*, as*X*varies; - the variance of the prediction based on
*X*, as*X*varies.

- the average of the variance of