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. The example below is for a continuous distribution over the interval 0<=x<=3.

Given the pdf we can find

To evaluate this expression we find

Henceto 3 dp.

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

These expressions are also useful for any continuous distribution defined on an interval