[PROPOSITIONS] Mass, centroids and probability

Normalising a density

If $\rho\ge0$ on $\Omega$ and

0<M=\iint_\Omega\rho(x,y)\,dA<\infty,

then

p(x,y)=\frac{\rho(x,y)}{M}

is a two-dimensional probability density on $\Omega$ , because

\iint_\Omega p(x,y)\,dA=1.

If $\rho$ is a population density, $\iint_R\rho\,dA$ is the population in a subregion $R\subseteq\Omega$ . The normalised density $p$ has units of inverse area, so $p\,dA$ is dimensionless.

After normalisation, the centre-of-mass coordinates become expected values:

\mathbb E[X]=\iint_\Omega xp(x,y)\,dA, \qquad \mathbb E[Y]=\iint_\Omega yp(x,y)\,dA.

For an event represented by a subregion $E\subseteq\Omega$ , the probability is

\mathbb P((X,Y)\in E)=\iint_E p(x,y)\,dA.

This is the population formula with the normalisation making the whole region have probability $1$ . Mass, a population centre and an expected value are all the same weighted-average construction.

[1]:Mass density of a lamina

Propositions from Mass, centroids and probability

Normalising a density

On this page