# What is CDF - Cumulative distribution function?

Could someone please explain in layman's terms what CDF is?

If someone could show a real-life example where this could be useful, it would be great.

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Interesting concept for a question. We can imagine more: "What's a real-life example where groups could be useful?" ... "What's a real-life example where linear independence could be useful?" ... "What's a real-life example where uncountable sets could be useful?" –  GEdgar Jul 19 '11 at 14:31

If you have a quantity $A$ that takes some value at random, the cumulative density function $F(x)$ gives the probability that $X$ is less than or equal to $x$, that is:

$$F(x) = P(A\leq x)$$

So you know several things: that $F(x)$ is bounded below by 0, and bounded above by 1 (because it doesn't make sense to have a probability outside [0,1]) and that it has to be increasing (or at least, non-decreasing) with $x$.

For example, if $A$ is the height of a person selected at random (imagine you knock on someone's door and measure the height of the person who answers) then $F(x)$ is the chance that the person will be shorter than $x$. Maybe $F(\textrm{180 cm}) = 0.8$, which means that there's an 80% chance that a person selected at random will be shorter than 180 cm (or equivalently, a 20% chance that they will be taller than 180cm).

A real-life example comes from finance. One way of measuring the risk of a portfolio (of stocks, for example) is to calculate the 5% daily value-at-risk, or VAR. To say that the 5% daily VAR is $x$ means you expect your loss to be worse than $x$ dollars on only 5% of days. For example, you might report that the 5% daily VAR is \$60,000, meaning that you expect to lose more than \$60,000 on 5% of days, and on the other 95% your loss will be less than \$60,000 (ideally, you will be in profit!) To calculate the 5% VAR we need to know the cumulative distribution function of our losses. If the cumulative distribution function of daily losses is$F(\cdot)$, then the 5% daily VAR is the value of$x\$ that solves the equation

$$F(x) = 0.05$$

The reporting of daily VAR is a requirement in financial institutions worldwide, so this certainly satisfies your requirements of a 'real-life' application!

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