From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- source/know/concept/random-variable/index.md | 146 +++++++++++++-------------- 1 file changed, 73 insertions(+), 73 deletions(-) (limited to 'source/know/concept/random-variable/index.md') diff --git a/source/know/concept/random-variable/index.md b/source/know/concept/random-variable/index.md index 1781c8b..ecb8e96 100644 --- a/source/know/concept/random-variable/index.md +++ b/source/know/concept/random-variable/index.md @@ -19,50 +19,50 @@ of a random variable. ## Probability space -A **probability space** or **probability triple** $(\Omega, \mathcal{F}, P)$ +A **probability space** or **probability triple** $$(\Omega, \mathcal{F}, P)$$ is the formal mathematical model of a given **stochastic experiment**, i.e. a process with a random outcome. -The **sample space** $\Omega$ is the set -of all possible outcomes $\omega$ of the experimement. -Those $\omega$ are selected randomly according to certain criteria. -A subset $A \subset \Omega$ is called an **event**, -and can be regarded as a true statement about all $\omega$ in that $A$. +The **sample space** $$\Omega$$ is the set +of all possible outcomes $$\omega$$ of the experimement. +Those $$\omega$$ are selected randomly according to certain criteria. +A subset $$A \subset \Omega$$ is called an **event**, +and can be regarded as a true statement about all $$\omega$$ in that $$A$$. -The **event space** $\mathcal{F}$ is a set of events $A$ +The **event space** $$\mathcal{F}$$ is a set of events $$A$$ that are interesting to us, -i.e. we have subjectively chosen $\mathcal{F}$ +i.e. we have subjectively chosen $$\mathcal{F}$$ based on the problem at hand. -Since events $A$ represent statements about outcomes $\omega$, +Since events $$A$$ represent statements about outcomes $$\omega$$, and we would like to use logic on those statemenets, -we demand that $\mathcal{F}$ is a [$\sigma$-algebra](/know/concept/sigma-algebra/). +we demand that $$\mathcal{F}$$ is a [$$\sigma$$-algebra](/know/concept/sigma-algebra/). -Finally, the **probability measure** or **probability function** $P$ -is a function that maps $A$ events to probabilities $P(A)$. -Formally, $P : \mathcal{F} \to \mathbb{R}$ is defined to satisfy: +Finally, the **probability measure** or **probability function** $$P$$ +is a function that maps $$A$$ events to probabilities $$P(A)$$. +Formally, $$P : \mathcal{F} \to \mathbb{R}$$ is defined to satisfy: -1. If $A \in \mathcal{F}$, then $P(A) \in [0, 1]$. -2. If $A, B \in \mathcal{F}$ do not overlap $A \cap B = \varnothing$, - then $P(A \cup B) = P(A) + P(B)$. -3. The total probability $P(\Omega) = 1$. +1. If $$A \in \mathcal{F}$$, then $$P(A) \in [0, 1]$$. +2. If $$A, B \in \mathcal{F}$$ do not overlap $$A \cap B = \varnothing$$, + then $$P(A \cup B) = P(A) + P(B)$$. +3. The total probability $$P(\Omega) = 1$$. -The reason we only assign probability to events $A$ -rather than individual outcomes $\omega$ is that -if $\Omega$ is continuous, all $\omega$ have zero probability, -while intervals $A$ can have nonzero probability. +The reason we only assign probability to events $$A$$ +rather than individual outcomes $$\omega$$ is that +if $$\Omega$$ is continuous, all $$\omega$$ have zero probability, +while intervals $$A$$ can have nonzero probability. ## Random variable -Once we have a probability space $(\Omega, \mathcal{F}, P)$, -we can define a **random variable** $X$ -as a function that maps outcomes $\omega$ +Once we have a probability space $$(\Omega, \mathcal{F}, P)$$, +we can define a **random variable** $$X$$ +as a function that maps outcomes $$\omega$$ to another set, usually the real numbers. To be a valid real-valued random variable, -a function $X : \Omega \to \mathbb{R}^n$ must satisfy the following condition, -in which case $X$ is said to be **measurable** -from $(\Omega, \mathcal{F})$ to $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$: +a function $$X : \Omega \to \mathbb{R}^n$$ must satisfy the following condition, +in which case $$X$$ is said to be **measurable** +from $$(\Omega, \mathcal{F})$$ to $$(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$$: $$\begin{aligned} \{ \omega \in \Omega : X(\omega) \in B \} \in \mathcal{F} @@ -70,16 +70,16 @@ $$\begin{aligned} \end{aligned}$$ In other words, for a given Borel set -(see [$\sigma$-algebra](/know/concept/sigma-algebra/)) $B \in \mathcal{B}(\mathbb{R}^n)$, -the set of all outcomes $\omega \in \Omega$ that satisfy $X(\omega) \in B$ -must form a valid event; this set must be in $\mathcal{F}$. +(see [$$\sigma$$-algebra](/know/concept/sigma-algebra/)) $$B \in \mathcal{B}(\mathbb{R}^n)$$, +the set of all outcomes $$\omega \in \Omega$$ that satisfy $$X(\omega) \in B$$ +must form a valid event; this set must be in $$\mathcal{F}$$. The point is that we need to be able to assign probabilities -to statements of the form $X \in [a, b]$ for all $a < b$, -which is only possible if that statement corresponds to an event in $\mathcal{F}$, -since $P$'s domain is $\mathcal{F}$. +to statements of the form $$X \in [a, b]$$ for all $$a < b$$, +which is only possible if that statement corresponds to an event in $$\mathcal{F}$$, +since $$P$$'s domain is $$\mathcal{F}$$. -Given such an $X$, and a set $B \subseteq \mathbb{R}$, -the **preimage** or **inverse image** $X^{-1}$ is defined as: +Given such an $$X$$, and a set $$B \subseteq \mathbb{R}$$, +the **preimage** or **inverse image** $$X^{-1}$$ is defined as: $$\begin{aligned} X^{-1}(B) @@ -87,16 +87,16 @@ $$\begin{aligned} \end{aligned}$$ As suggested by the notation, -$X^{-1}$ can be regarded as the inverse of $X$: -it maps $B$ to the event for which $X \in B$. -With this, our earlier requirement that $X$ be measurable -can be written as: $X^{-1}(B) \in \mathcal{F}$ for any $B \in \mathcal{B}(\mathbb{R}^n)$. -This is also often stated as "$X$ is *$\mathcal{F}$-measurable"*. - -Related to $\mathcal{F}$ is the **information** -obtained by observing a random variable $X$. -Let $\sigma(X)$ be the information generated by observing $X$, -i.e. the events whose occurrence can be deduced from the value of $X$, +$$X^{-1}$$ can be regarded as the inverse of $$X$$: +it maps $$B$$ to the event for which $$X \in B$$. +With this, our earlier requirement that $$X$$ be measurable +can be written as: $$X^{-1}(B) \in \mathcal{F}$$ for any $$B \in \mathcal{B}(\mathbb{R}^n)$$. +This is also often stated as "$$X$$ is *$$\mathcal{F}$$-measurable"*. + +Related to $$\mathcal{F}$$ is the **information** +obtained by observing a random variable $$X$$. +Let $$\sigma(X)$$ be the information generated by observing $$X$$, +i.e. the events whose occurrence can be deduced from the value of $$X$$, or, more formally: $$\begin{aligned} @@ -105,29 +105,29 @@ $$\begin{aligned} = \{ A \in \mathcal{F} : A = X^{-1}(B) \mathrm{\:for\:some\:} B \in \mathcal{B}(\mathbb{R}^n) \} \end{aligned}$$ -In other words, if the realized value of $X$ is -found to be in a certain Borel set $B \in \mathcal{B}(\mathbb{R}^n)$, -then the preimage $X^{-1}(B)$ (i.e. the event yielding this $B$) +In other words, if the realized value of $$X$$ is +found to be in a certain Borel set $$B \in \mathcal{B}(\mathbb{R}^n)$$, +then the preimage $$X^{-1}(B)$$ (i.e. the event yielding this $$B$$) is known to have occurred. -In general, given any $\sigma$-algebra $\mathcal{H}$, -a variable $Y$ is said to be *"$\mathcal{H}$-measurable"* -if $\sigma(Y) \subseteq \mathcal{H}$, -so that $\mathcal{H}$ contains at least -all information extractable from $Y$. +In general, given any $$\sigma$$-algebra $$\mathcal{H}$$, +a variable $$Y$$ is said to be *"$$\mathcal{H}$$-measurable"* +if $$\sigma(Y) \subseteq \mathcal{H}$$, +so that $$\mathcal{H}$$ contains at least +all information extractable from $$Y$$. -Note that $\mathcal{H}$ can be generated by another random variable $X$, -i.e. $\mathcal{H} = \sigma(X)$. +Note that $$\mathcal{H}$$ can be generated by another random variable $$X$$, +i.e. $$\mathcal{H} = \sigma(X)$$. In that case, the **Doob-Dynkin lemma** states -that $Y$ is only $\sigma(X)$-measurable -if $Y$ can always be computed from $X$, -i.e. there exists a function $f$ such that -$Y(\omega) = f(X(\omega))$ for all $\omega \in \Omega$. +that $$Y$$ is only $$\sigma(X)$$-measurable +if $$Y$$ can always be computed from $$X$$, +i.e. there exists a function $$f$$ such that +$$Y(\omega) = f(X(\omega))$$ for all $$\omega \in \Omega$$. Now, we are ready to define some familiar concepts from probability theory. -The **cumulative distribution function** $F_X(x)$ is -the probability of the event where the realized value of $X$ -is smaller than some given $x \in \mathbb{R}$: +The **cumulative distribution function** $$F_X(x)$$ is +the probability of the event where the realized value of $$X$$ +is smaller than some given $$x \in \mathbb{R}$$: $$\begin{aligned} F_X(x) @@ -136,8 +136,8 @@ $$\begin{aligned} = P(X^{-1}(]\!-\!\infty, x])) \end{aligned}$$ -If $F_X(x)$ is differentiable, -then the **probability density function** $f_X(x)$ is defined as: +If $$F_X(x)$$ is differentiable, +then the **probability density function** $$f_X(x)$$ is defined as: $$\begin{aligned} f_X(x) @@ -147,10 +147,10 @@ $$\begin{aligned} ## Expectation value -The **expectation value** $\mathbf{E}[X]$ of a random variable $X$ +The **expectation value** $$\mathbf{E}[X]$$ of a random variable $$X$$ can be defined in the familiar way, as the sum/integral -of every possible value of $X$ mutliplied by the corresponding probability (density). -For continuous and discrete sample spaces $\Omega$, respectively: +of every possible value of $$X$$ mutliplied by the corresponding probability (density). +For continuous and discrete sample spaces $$\Omega$$, respectively: $$\begin{aligned} \mathbf{E}[X] @@ -160,18 +160,18 @@ $$\begin{aligned} = \sum_{i = 1}^N x_i \: P(X \!=\! x_i) \end{aligned}$$ -However, $f_X(x)$ is not guaranteed to exist, +However, $$f_X(x)$$ is not guaranteed to exist, and the distinction between continuous and discrete is cumbersome. -A more general definition of $\mathbf{E}[X]$ +A more general definition of $$\mathbf{E}[X]$$ is the following Lebesgue-Stieltjes integral, -since $F_X(x)$ always exists: +since $$F_X(x)$$ always exists: $$\begin{aligned} \mathbf{E}[X] = \int_{-\infty}^\infty x \dd{F_X(x)} \end{aligned}$$ -This is valid for any sample space $\Omega$. +This is valid for any sample space $$\Omega$$. Or, equivalently, a Lebesgue integral can be used: $$\begin{aligned} @@ -182,8 +182,8 @@ $$\begin{aligned} An expectation value defined in this way has many useful properties, most notably linearity. -We can also define the familiar **variance** $\mathbf{V}[X]$ -of a random variable $X$ as follows: +We can also define the familiar **variance** $$\mathbf{V}[X]$$ +of a random variable $$X$$ as follows: $$\begin{aligned} \mathbf{V}[X] -- cgit v1.2.3