Summary table of probability terms

This page is a summary table of probability terms.

Table

Term	Symbol	Type	Definition
Reals	$\mathbf {R}$
Borel subsets of the reals	${\mathcal {B}}$
Sample space	$\Omega$
Outcome	$\omega$	$\Omega$
Events or measurable sets	${\mathcal {F}}$
Probability measure	$\mathbf {P}$ or $\Pr$ or $\mathbf {P} _{\mathcal {F}}$	${\mathcal {F}}\to [0,1]$
Probability triple or probability space	$(\Omega ,{\mathcal {F}},\mathbf {P} )$
Distribution	$\mu$ or ${\mathcal {D}}$ or $D$ or $\mathbf {P} _{\mathcal {B}}$ or ${\mathcal {L}}(X)$ or $\mathbf {P} X^{-1}$	${\mathcal {B}}\to \mathbf {[} 0,1]$	$B\mapsto \mathbf {P} (X\in B)$
Induced probability space	$(\mathbf {R} ,{\mathcal {B}},\mu )$
Cumulative distribution function or CDF	$F_{X}$	$\mathbf {R} \to [0,1]$
Probability density function or PDF	$f_{X}$	$\mathbf {R} \to [0,\infty )$
Random variable	$X$	$\Omega \to \mathbf {R}$
Indicator of $A$	$1_{A}$	$\Omega \to \{0,1\}$
Expectation	$\mathbf {E}$ or $\mathrm {E}$	$(\Omega \to \mathbf {R} )\to \mathbf {R}$

Let $(\Omega ,{\mathcal {F}},\mathbf {P} )$ be a probability space.

Given a random variable $X$ , we can compute its distribution $\mu$ . How? Just let $\mu (B)=\mathbf {P} _{\mathcal {F}}(X\in B)$
Given a random variable, we can compute the probability density function. How?
Given a random variable, we can compute the cumulative distribution function. How?
Given a distribution, we can retrieve a random variable. But this random variable is not unique? This is why we can say stuff like "let $X\sim {\mathcal {D}}$ ".
Given a distribution $\mu$ , we can compute its density function. How? Just find the derivative of $\mu ((-\infty ,x])$ . (?)
Given a cumulative distribution function, we can compute the random variable. (Right?)
Given a probability density function, can we get everything else? Don't we just have to integrate to get the cdf, which gets us the random variable and the distribution?
Given a cumulative distribution function, how do we get the distribution? We have $F_{X}(x)=\mathbf {P} _{\mathcal {F}}(X\leq x)=\mathbf {P} _{\mathcal {B}}((-\infty ,x])$ , which gets us some of what the distribution $\mathbf {P} _{\mathcal {B}}$ maps to, but ${\mathcal {B}}$ is bigger than this. What do we do about the other value we need to map? We can compute intervals like $F_{X}(b)-F_{X}(a)=\mathbf {P} _{\mathcal {F}}(a\leq X\leq b)=\mathbf {P} _{\mathcal {B}}([a,b])$ .