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	$\mathrm{\Omega }$
Outcome	$\omega$	$\mathrm{\Omega }$
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	$(\mathrm{\Omega },\mathcal{F},\mathbf{P})$
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	$f_X$	$\mathbf{R}\to [0,\mathrm{\infty })$
Random variable	$X$	$\mathrm{\Omega }\to \mathbf{R}$
Indicator of $A$	$1_A$	$\mathrm{\Omega }\to \{0,1\}$
Expectation	$\mathbf{E}$ or $\mathrm{E}$	$(\mathrm{\Omega }\to \mathbf{R})\to \mathbf{R}$

Dependencies

Let $(\mathrm{\Omega },\mathcal{F},\mathbf{P})$ 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 $\mu ((-\mathrm{\infty },x])$. (?)
• 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 $F_X(x)={\mathbf{P}}_{\mathcal{F}}(X\le x)={\mathbf{P}}_{\mathcal{B}}((-\mathrm{\infty },x])$, 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 values we need to map? We can compute intervals like $F_X(b)-F_X(a)={\mathbf{P}}_{\mathcal{F}}(a\le X\le b)={\mathbf{P}}_{\mathcal{B}}([a,b])$. And we can apparently do the same for unions and limiting operations.

Philosophical details about the sample space

Given a random variable $X:\mathrm{\Omega }\to \mathbf{R}$ and any reasonable predicate $P$ about $X$, we can replace $P(X)$ with its extension $\{\omega \in \mathrm{\Omega }:P(X(\omega ))\}=\{\omega \in \mathrm{\Omega }:X(\omega )\in B\}$ for some $B\in \mathcal{B}$. And from then on, we can write ${\mathbf{P}}_{\mathcal{F}}(X\in B)$ as ${\mathbf{P}}_{\mathcal{B}}(B)=\mu (B)$.

See also

• Comparison of machine learning textbooks

External links

• 254A, Notes 0: A review of probability theory by Terence Tao
• Basic Random Variable Concepts by Kenneth Kreutz-Delgado
• Various questions on Mathematics Stack Exchange:
  • https://math.stackexchange.com/questions/2233731/discarding-random-variables-in-favor-of-a-domain-less-definition
  • https://math.stackexchange.com/questions/18198/what-are-the-sample-spaces-when-talking-about-continuous-random-variables
  • https://math.stackexchange.com/questions/2233721/the-true-domain-of-random-variables
  • https://math.stackexchange.com/questions/712734/domain-of-a-random-variable-sample-space-or-probability-space
  • https://math.stackexchange.com/questions/23006/the-role-of-the-hidden-probability-space-on-which-random-variables-are-defined
  • https://math.stackexchange.com/questions/1612012/how-should-i-understand-the-probability-space-omega-mathcalf-p-what-d
  • https://math.stackexchange.com/questions/2531810/why-does-probability-theory-insist-on-sample-spaces