Continuous distribution

Definition taught in Random variables and distributions, part of Probability

Continuous distribution

Densities of three exponential laws, with rates 0.5, 1 and 2, and of three normal laws. From Random variables and distributions.
Type Definition
Taught in Random variables and distributions
Part Probability
Worked examples 2

X is continuous when its distribution function can be written F(x)=\int_{-\infty}^{x} f(u)\,\,\mathrm{d} u for a non-negative function f with \int_{-\infty}^{\infty} f=1. The function f is a probability density. Then \mathbb{P}(X\in B)=\int_B f, and f=F' at every point where F is differentiable.

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.

  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.

A point drawn from the unit interval

Question page: A point drawn from the unit interval

A point is drawn from [0,1] so that probabilities are lengths.

  1. Find the probability that the point is less than 1/2.

  2. Find the probability that it lies between 1/3 and 3/4.

  3. The draw is repeated independently. Find the probability that the sum of the two points exceeds 1/2.

  1. The length of [0,1/2) is 1/2.

  2. The length of [1/3,3/4] is 3/4-1/3=5/12.

  3. The two draws are uniform on the unit square. The corner \{(x,y):x\ge 0,\,y\ge 0,\,x+y\le 1/2\} is a right triangle of area (1/2)\times(1/2)\times(1/2)=1/8. The complementary probability is 1-1/8=7/8.

Figures

Mass, density and distribution function. Left: a fair die’s mass function, and its distribution function, a right-continuous staircase whose jumps are the masses. Right: an \operatorname{Exp}(1) density, with \mathbb{P}(0.5<X\le1.6) as the area under it, and the same probability as the rise F(1.6)-F(0.5) of the distribution function. From Random variables and distributions.

Used in

See also

Cumulative distribution function · Likelihood function · Random variable · Discrete distribution · Expectation · Expectation of a function · Variance · Probability generating function

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