Applications of Calculus

The area under the probability distribution is of course an integral. If we call the random number x and the probability distribution D(x), then the probability that x lies in a certain range is given by

What about averages? If x had a finite number of equally probable values, we would simply add them up and divide by how many we had. If they weren't equally likely, we'd make the weighted average x1P1 + x2P2+... But we need to generalize this to a variable x that can take on any of a continuum of values. The continuous version of a sum is an integral, so the average is

where the integral is over all possible values of x.

Probability distribution for radioactive decay

Average lifetime