From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/random-variable/index.md | 202 +++++++++++++++++++++++++++ 1 file changed, 202 insertions(+) create mode 100644 source/know/concept/random-variable/index.md (limited to 'source/know/concept/random-variable') diff --git a/source/know/concept/random-variable/index.md b/source/know/concept/random-variable/index.md new file mode 100644 index 0000000..7243d15 --- /dev/null +++ b/source/know/concept/random-variable/index.md @@ -0,0 +1,202 @@ +--- +title: "Random variable" +date: 2021-10-22 +categories: +- Mathematics +- Statistics +- Measure theory +layout: "concept" +--- + +**Random variables** are the bread and butter +of probability theory and statistics, +and are simply variables whose value depends +on the outcome of a random experiment. +Here, we will describe the formal mathematical definition +of a random variable. + + +## Probability space + +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 **event space** $\mathcal{F}$ is a set of events $A$ +that are interesting to us, +i.e. we have subjectively chosen $\mathcal{F}$ +based on the problem at hand. +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/). + +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 $P(\Omega) = 1$. + +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 $\omega$ +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 $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$: + +$$\begin{aligned} + \{ \omega \in \Omega : X(\omega) \in B \} \in \mathcal{F} + \quad \mathrm{for\:any\:} B \in \mathcal{B}(\mathbb{R}^n) +\end{aligned}$$ + +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 $\mathcal{F}$. +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 $\mathcal{F}$. + +Given such an $X$, and a set $B \subseteq \mathbb{R}$, +the **preimage** or **inverse image** $X^{-1}$ is defined as: + +$$\begin{aligned} + X^{-1}(B) + = \{ \omega \in \Omega : X(\omega) \in B \} +\end{aligned}$$ + +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 $X$, +or, more formally: + +$$\begin{aligned} + \sigma(X) + = X^{-1}(\mathcal{B}(\mathbb{R}^n)) + = \{ A \in \mathcal{F} : A = X^{-1}(B) \mathrm{\:for\:some\:} B \in \mathcal{B}(\mathbb{R}^n) \} +\end{aligned}$$ + +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 $B$) +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 $Y$. + +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 $\omega \in \Omega$. + +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 $x \in \mathbb{R}$: + +$$\begin{aligned} + F_X(x) + = P(X \le x) + = P(\{ \omega \in \Omega : X(\omega) \le x \}) + = P(X^{-1}(]\!-\!\infty, x])) +\end{aligned}$$ + +If $F_X(x)$ is differentiable, +then the **probability density function** $f_X(x)$ is defined as: + +$$\begin{aligned} + f_X(x) + = \dv{F_X}{x} +\end{aligned}$$ + + +## Expectation value + +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: + +$$\begin{aligned} + \mathbf{E}[X] + = \int_{-\infty}^\infty x \: f_X(x) \dd{x} + \qquad \mathrm{or} \qquad + \mathbf{E}[X] + = \sum_{i = 1}^N x_i \: P(X \!=\! x_i) +\end{aligned}$$ + +However, $f_X(x)$ is not guaranteed to exist, +and the distinction between continuous and discrete is cumbersome. +A more general definition of $\mathbf{E}[X]$ +is the following Lebesgue-Stieltjes integral, +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 $\Omega$. +Or, equivalently, a Lebesgue integral can be used: + +$$\begin{aligned} + \mathbf{E}[X] + = \int_\Omega X(\omega) \dd{P(\omega)} +\end{aligned}$$ + +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: + +$$\begin{aligned} + \mathbf{V}[X] + = \mathbf{E}\big[ (X - \mathbf{E}[X])^2 \big] + = \mathbf{E}[X^2] - \big(\mathbf{E}[X]\big)^2 +\end{aligned}$$ + +It is also possible to calculate expectation values and variances +adjusted to some given event information: +see [conditional expectation](/know/concept/conditional-expectation/). + + + +## References +1. U.H. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. -- cgit v1.2.3