Summary table of probability terms

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	$F_X$	$\mathbf{R}\to \mathbf{R}$
Probability density function or PDF	$f_X$	$\mathbf{R}\to \mathbf{R}$
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, we can compute its distribution.
• Given a random variable, we can compute the probability density function.
• Given a random variable, we can compute the cumulative distribution function.
• Given a distribution, we can retrieve the random variable. (Right?) This is why we can say stuff like "let $X\sim \mathcal{D}$".
• 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?

See also

• Comparison of machine learning textbooks

External links

• 254A, Notes 0: A review of probability theory by wikipedia:Terence Tao