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Diffstat (limited to 'content/know/concept/random-variable/index.pdc')
| -rw-r--r-- | content/know/concept/random-variable/index.pdc | 40 |
1 files changed, 38 insertions, 2 deletions
diff --git a/content/know/concept/random-variable/index.pdc b/content/know/concept/random-variable/index.pdc index 2a8643e..bc41744 100644 --- a/content/know/concept/random-variable/index.pdc +++ b/content/know/concept/random-variable/index.pdc @@ -73,7 +73,8 @@ $$\begin{aligned} \quad \mathrm{for\:any\:} B \in \mathcal{B}(\mathbb{R}^n) \end{aligned}$$ -In other words, for a given Borel set (see $\sigma$-algebra) $B \in \mathcal{B}(\mathbb{R}^n)$, +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 @@ -94,7 +95,38 @@ $X^{-1}$ 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"*. +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 @@ -163,6 +195,10 @@ $$\begin{aligned} = \mathbf{E}[X^2] - \big(\mathbf{E}[X]\big)^2 \end{aligned}$$ +It is also possible to calculate expectation values and variances +adjusted to some given event information: +see [conditional expectation](/know/concept/conditional-expectation/). + ## References |