# Why do we need to estimate the cumulative distribution function?

In statistic estimation we always works with densities. However, sometimes we need to estimate the cumulative distribution function? why? What is the benefits of estimating the cumulative distribution function? Every time that I search about cumulative distribution function or read a book, I found a mathematical expression only!! I really need to understand why we need them, in which cases that they are help?

• (1) What's the problem with a "mathematical expression only"? Couldn't you level the same criticism at literally every mathematical object used in statistics, including density and probability functions? (2) Do you have any interest in modeling discrete variables like counts, occurrences, and categories? Their distributions never have densities. – whuber Jan 11 '18 at 14:09
• @whuber The problem is I would like to understand when I use distribution and when I cannot. Or, when the distribution is needed in modeling and when the density is needed? – Silver_80 Jan 12 '18 at 10:28
• That's sort of like asking for which math problems addition is called for and which ones need you to multiply. It all depends on the problem. – whuber Jan 12 '18 at 14:52