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. 
, we
readily
obtain the following expressions for the mean and variance of 
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.
