We can define the expectation value and variance for a continuous pdf, consistent with our earlier definitions for a discrete pdf:

Similarly, if we define a real-valued function of the r.v. **x**, we
readily
obtain the following expressions for the mean and variance of **g** for a
continuous pdf:

It is important to keep in mind that the quantities and
are * true* means,
* properties* of the pdf and the function
In many cases of practical interest the true mean is not known, and the
purpose of the Monte Carlo simulation will be to * estimate* the true
mean.
These estimates will be denoted by
a caret or hat, e.g., and .
Thus the result of a Monte Carlo simulation
might be , and the hope is that this is a good approximation to the
true (but unknown) quantity .
This notation will be adhered to throughout this chapter on Monte Carlo
methods.