Every probability distribution has a probability density function, in terms of which it is usually defined. The probability density function for the normal distribution isfor the uniform distribution it isfor values betweenandand zero outside this interval. Both of these examples are continuous distributions, where any value in an interval may occur, though there are many examples of normal and uniform discrete distributions. Some distributions are defined by the values they take over an interval, either one by one in the case of a discrete distribution, or in terms of a function, for either continuous or discrete distributions. For example:

Given the pdf we can find

To evaluate this expression we find

Hence

The expression for the varianceis fundamental in higher mathematics and physics, especially quantum physics.