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+---
+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.