Atom of a distribution
Topic of Random variables and laws, part of Stochastic processes
Atom of a distribution
| Type | Topic |
| Taught in | Random variables and laws |
| Part | Stochastic processes |
A point x with \mathrm{P}(X=x)>0. The distribution function jumps there by exactly that probability, F_X(x)-F_X(x-), and right-continuity puts the atom in F_X(x)=\mathrm{P}(X\le x).
In Random variables and laws
The distribution function is non-decreasing and right-continuous, with limit 0 at -\infty and limit 1 at +\infty. Right-continuity is the convention F_X(x)=\mathrm{P}(X\le x): the value at x includes the atom at x. The jump at x equals that atom,
\mathrm{P}(X=x)=F_X(x)-F_X(x-).
Used in
See also
Cumulative distribution function · Measurable random variable · Simple variables and expectation · Lebesgue decomposition · Kolmogorov extension theorem