4.2 Probability and Centers of Gravity

Definition 4.2.1 (Center of gravity of a body).

1. If a body $A\subset \R^n$ is made of some homogenous material, then the center of gravity of $A$ is the point $\bar x$ whose $i$th coordinate is

$$ \bar x_i = \frac{\int_Ax_i|d^nx|}{\int_A|d^nx|}. $$

2. If $A$ has variable density given by the function $\mu: A\to \R$, then the mass $M(A)$ of such body is

$$ M(A) = \int_A\mu(x)|d^nx| $$

and the center of gravity is

$$ \bar x_i = \frac{\int_Ax_i\mu(x)|d^nx|}{M(A)}. $$

Definition 4.2.3 (Probability density). Let $S, P$ be a probability space, with $S\subset \R^n$. If there exists a nonnegative integrable function $\mu : \R^n \to \R$ such that event $A$

$$ P(A) = \int_A\mu(x)|d^nx|, $$

then $\mu$ is the probability density of $S$, $P$. The probability density must staisfy

$$ \mu(x)\geq 0 \text{ and }\int_{\R^k}\mu(x)|d^kx|=1. $$

Definition 4.2.6 (Expected value). Let $S\subset \R^n$ be a sample space with probability density $\mu$. If $f$ be a random variable such that $f(x)\mu(x)$ is integrable, the expected vlaue of $f$ is

$$ E(f) = \int_Sf(x)\mu(x)|d^nx|. $$

Note that this is the same as the center of gravity, except that the mass is 1.

Central Limit Theorem

The normal or Gaussian distribution is one of the most ubiquitous in probability theory. In one variable, it is

$$ \mu(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}. $$