TITLE: Why is there a preference to use the cumulative distribution function to characterise a random variable instead of the probability density function?
QUESTION [17 upvotes]: Perhaps this is but a subtlety but I've noticed that in quite a few definitions in statistics and probability, definitions regarding the distribution of a variable or a sample of data choose to use the cumulative distribution function to characterise random variables as following a specific distribution as opposed to using the probability density function.
E.g with the Kolmogorov Smirnov test, we look at the difference between cdfs and not pdfs.
Is there a specific reason for this ?

REPLY [15 votes]: You pretty much always want to calculate Pr(a < x < b), which is a simple subtraction if you have a CDF. If you're using a PDF, you have to integrate, which is a very complex operation.