Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
Home Modern Physics Rules of Randomness Applications of Calculus  
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Applications of CalculusThe 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.


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