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.