Cumulative distribution function

Definition taught in Random variables and distributions, part of Probability

Cumulative distribution function

Mass, density and distribution function. From Random variables and distributions.

Type	Definition
Taught in	Random variables and distributions
Part	Probability
Properties	1
Worked examples	2
Questions using it	7

The cumulative distribution function of X is F(x)=\mathbb{P}(X\le x), for x\in\mathbb{R}.

Properties

Inverse-transform method

Main article: Inverse-transform method

Let F be a distribution function and define F^{-1}(u)=\inf\{x:F(x)\ge u\} for u\in(0,1). If U\sim\operatorname{U}[0,1], then X=F^{-1}(U) has distribution function F.

Examples

A cubic distribution function

Question page: A cubic distribution function

A continuous random variable X has distribution function F(x)=0 for x<0, F(x)=x^3 for x\in[0,1], and F(x)=1 for x>1.

1. Find the density of X.

2. Calculate \mathbb{P}(1/2<X<3/4).

3. Calculate \mathbb{E} X.

Worked answer (click to show)

1. f(x)=F'(x)=3x^2 for 0<x<1, and f(x)=0 elsewhere.

2. \mathbb{P}(1/2<X<3/4)=F(3/4)-F(1/2)=(3/4)^3-(1/2)^3=19/64.

3. \mathbb{E} X=\int_0^1 x\cdot 3x^2\,\,\mathrm{d} x=3/4.

Distribution function of a maximum

Question page: Distribution function of a maximum

Let X_1,\ldots,X_n be independent with distribution functions F_1,\ldots,F_n. Express the distribution function of M=\max(X_1,\ldots,X_n) in terms of F_1,\ldots,F_n.

Worked answer (click to show)

The maximum is at most m when every variable is at most m. Independence factors the probability:

\mathbb{P}(M\le m)=F_1(m)\cdots F_n(m).

Figures

The inverse-transform method. A uniform draw U on the vertical axis is carried across to the graph of F and down to the horizontal axis, giving X=F^{-1}(U). Steep parts of F catch many values of U, so X lands there often: that is where the density is large. From Simulation.

Used in

• Taught in Random variables and distributions, where the derivation and examples are.
• Also used in Simulation · Joint distributions · Functions of random variables and limit theorems · Reliability · Using R
• Questions that use it (7) — The larger coordinate · A cubic distribution function · Draws from an exponential law (answer) · A shifted exponential density (answer) · Minimum and average of uniform draws (answer) · Distribution function of a maximum · A normal law with mean 2
• All 27 questions on Random variables and distributions

See also

Central limit theorem · Continuous distribution · Exponential distribution · Inverse-transform method · Normal distribution · Survival function · Weibull distribution · Random variable

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