Cumulative distribution function

Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function (ccdf) or simply the tail distribution or exceedance, and is defined as

$${\displaystyle {\bar {F}}_{X}(x)=\operatorname {P} (X>x)=1-F_{X}(x).}$$

This has applications in statistical hypothesis testing, for example, because the one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic, T, has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value ${\displaystyle t}$ of the test statistic

$${\displaystyle p=\operatorname {P} (T\geq t)=\operatorname {P} (T>t)=1-F_{T}(t).}$$

In survival analysis, ${\displaystyle {\bar {F}}_{X}(x)}$ is called the survival function and denoted ${\displaystyle S(x)}$, while the term reliability function is common in engineering.

Properties

• For a non-negative continuous random variable having an expectation, Markov's inequality states that[4]
  $${\displaystyle {\bar {F}}_{X}(x)\leq {\frac {\operatorname {E} (X)}{x}}.}$$

  
• As ${\displaystyle x\to \infty ,{\bar {F}}_{X}(x)\to 0}$, and in fact ${\displaystyle {\bar {F}}_{X}(x)=o(1/x)}$ provided that ${\displaystyle \operatorname {E} (X)}$ is finite.
  Proof:
  Assuming ${\displaystyle X}$ has a density function ${\displaystyle f_{X}}$, for any ${\displaystyle c>0}$
  $${\displaystyle \operatorname {E} (X)=\int _{0}^{\infty }xf_{X}(x)\,dx\geq \int _{0}^{c}xf_{X}(x)\,dx+c\int _{c}^{\infty }f_{X}(x)\,dx}$$

  
  Then, on recognizing
  $${\displaystyle {\bar {F}}_{X}(c)=\int _{c}^{\infty }f_{X}(x)\,dx}$$

  
  and rearranging terms,
  $${\displaystyle 0\leq c{\bar {F}}_{X}(c)\leq \operatorname {E} (X)-\int _{0}^{c}xf_{X}(x)\,dx\to 0{\text{ as }}c\to \infty }$$

  
  as claimed.
• For a random variable having an expectation,
  $${\displaystyle \operatorname {E} (X)=\int _{0}^{\infty }{\bar {F}}_{X}(x)\,dx-\int _{-\infty }^{0}F_{X}(x)\,dx}$$

  
  and for a non-negative random variable the second term is 0.
  If the random variable can only take non-negative integer values, this is equivalent to
  $${\displaystyle \operatorname {E} (X)=\sum _{n=0}^{\infty }{\bar {F}}_{X}(n).}$$

  

Example of the folded cumulative distribution for a normal distribution function with an expected value of 0 and a standard deviation of 1.

While the plot of a cumulative distribution ${\displaystyle F}$ often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over,[5][6] that is

${\displaystyle F_{\text{fold}}(x)=F(x)1_{\{F(x)\leq 0.5\}}+(1-F(x))1_{\{F(x)>0.5\}}}$

where ${\displaystyle 1_{\{A\}}}$ denotes the indicator function and the second summand is the survivor function, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median, dispersion (specifically, the mean absolute deviation from the median[7]) and skewness of the distribution or of the empirical results.

If the CDF F is strictly increasing and continuous then ${\displaystyle F^{-1}(p),p\in [0,1],}$ is the unique real number ${\displaystyle x}$ such that ${\displaystyle F(x)=p}$. This defines the inverse distribution function or quantile function.

Some distributions do not have a unique inverse (for example if ${\displaystyle f_{X}(x)=0}$ for all ${\displaystyle a<x<b}$, causing ${\displaystyle F_{X}}$ to be constant). In this case, one may use the generalized inverse distribution function, which is defined as

${\displaystyle F^{-1}(p)=\inf\{x\in \mathbb {R} :F(x)\geq p\},\quad \forall p\in [0,1].}$

• Example 1: The median is ${\displaystyle F^{-1}(0.5)}$.
• Example 2: Put ${\displaystyle \tau =F^{-1}(0.95)}$. Then we call ${\displaystyle \tau }$ the 95th percentile.

Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:

1. ${\displaystyle F^{-1}}$ is nondecreasing[8]
2. ${\displaystyle F^{-1}(F(x))\leq x}$
3. ${\displaystyle F(F^{-1}(p))\geq p}$
4. ${\displaystyle F^{-1}(p)\leq x}$ if and only if ${\displaystyle p\leq F(x)}$
5. If ${\displaystyle Y}$ has a ${\displaystyle U[0,1]}$ distribution then ${\displaystyle F^{-1}(Y)}$ is distributed as ${\displaystyle F}$. This is used in random number generation using the inverse transform sampling-method.
6. If ${\displaystyle \{X_{\alpha }\}}$ is a collection of independent ${\displaystyle F}$-distributed random variables defined on the same sample space, then there exist random variables ${\displaystyle Y_{\alpha }}$ such that ${\displaystyle Y_{\alpha }}$ is distributed as ${\displaystyle U[0,1]}$ and ${\displaystyle F^{-1}(Y_{\alpha })=X_{\alpha }}$ with probability 1 for all ${\displaystyle \alpha }$.

The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.

The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function.[9]

When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables ${\displaystyle X,Y}$, the joint CDF ${\displaystyle F_{XY}}$ is given by[2]^{: p. 89}

$${\displaystyle F_{X,Y}(x,y)=\operatorname {P} (X\leq x,Y\leq y)}$$

(Eq.3)

where the right-hand side represents the probability that the random variable ${\displaystyle X}$ takes on a value less than or equal to ${\displaystyle x}$ and that ${\displaystyle Y}$ takes on a value less than or equal to ${\displaystyle y}$.

Example of joint cumulative distribution function:

For two continuous variables X and Y:

$${\displaystyle \Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;}$$

For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example:[10]

given the joint probability mass function in tabular form, determine the joint cumulative distribution function.

Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form:

For ${\displaystyle N}$ random variables ${\displaystyle X_{1},\ldots ,X_{N}}$, the joint CDF ${\displaystyle F_{X_{1},\ldots ,X_{N}}}$ is given by

$${\displaystyle F_{X_{1},\ldots ,X_{N}}(x_{1},\ldots ,x_{N})=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}$$

(Eq.4)

Interpreting the ${\displaystyle N}$ random variables as a random vector ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}}$ yields a shorter notation:

$${\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}$$

Every multivariate CDF is:

1. Monotonically non-decreasing for each of its variables,
2. Right-continuous in each of its variables,
3. ${\displaystyle 0\leq F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})\leq 1,}$
4. $${\displaystyle \lim _{x_{1},\ldots ,x_{n}\rightarrow +\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=1{\text{ and }}\lim _{x_{i}\rightarrow -\infty }F_{X_{1}\ldots X_{n}}(x_{1},\ldots ,x_{n})=0,{\text{for all }}i.}$$

   

Not every function satisfying the above four properties is a multivariate CDF, unlike in the single dimension case. For example, let ${\displaystyle F(x,y)=0}$ for ${\displaystyle x<0}$ or ${\displaystyle x+y<1}$ or ${\displaystyle y<0}$ and let ${\displaystyle F(x,y)=1}$ otherwise. It is easy to see that the above conditions are met, and yet ${\displaystyle F}$ is not a CDF since if it was, then ${\textstyle \operatorname {P} \left({\frac {1}{3}}<X\leq 1,{\frac {1}{3}}<Y\leq 1\right)=-1}$ as explained below.

The probability that a point belongs to a hyperrectangle is analogous to the 1-dimensional case:[11]

$${\displaystyle F_{X_{1},X_{2}}(a,c)+F_{X_{1},X_{2}}(b,d)-F_{X_{1},X_{2}}(a,d)-F_{X_{1},X_{2}}(b,c)=\operatorname {P} (a<X_{1}\leq b,c<X_{2}\leq d)=\int ...}$$

The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form ${\displaystyle P(Z\leq 1+2i)}$ make no sense. However expressions of the form ${\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)}$ make sense. Therefore, we define the cumulative distribution of a complex random variables via the joint distribution of their real and imaginary parts:

$${\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)}).}$$

Generalization of Eq.4 yields

$${\displaystyle F_{\mathbf {Z} }(\mathbf {z} )=F_{\Re {(Z_{1})},\Im {(Z_{1})},\ldots ,\Re {(Z_{n})},\Im {(Z_{n})}}(\Re {(z_{1})},\Im {(z_{1})},\ldots ,\Re {(z_{n})},\Im {(z_{n})})=\operatorname {P} (\Re {(Z_{1})}\leq \Re {(z_{1})},\Im {(Z_{1})}\leq \Im {(z_{1})},\ldots ,\Re {(Z_{n})}\leq \Re {(z_{n})},\Im {(Z_{n})}\leq \Im {(z_{n})})}$$

as definition for the CDS of a complex random vector ${\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{N})^{T}}$.

The Kolmogorov–Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper's test is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.