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Diffstat (limited to 'source/know/concept/random-variable/index.md')
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1 files changed, 10 insertions, 7 deletions
diff --git a/source/know/concept/random-variable/index.md b/source/know/concept/random-variable/index.md index ecb8e96..a6cbc8b 100644 --- a/source/know/concept/random-variable/index.md +++ b/source/know/concept/random-variable/index.md @@ -17,6 +17,7 @@ Here, we will describe the formal mathematical definition of a random variable. + ## Probability space A **probability space** or **probability triple** $$(\Omega, \mathcal{F}, P)$$ @@ -24,7 +25,7 @@ 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. +of all possible outcomes $$\omega$$ of the stochastic experiment. 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 $$A$$. @@ -34,7 +35,7 @@ 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, +and we would like to use logic on those statements, we demand that $$\mathcal{F}$$ is a [$$\sigma$$-algebra](/know/concept/sigma-algebra/). Finally, the **probability measure** or **probability function** $$P$$ @@ -52,6 +53,7 @@ if $$\Omega$$ is continuous, all $$\omega$$ have zero probability, while intervals $$A$$ can have nonzero probability. + ## Random variable Once we have a probability space $$(\Omega, \mathcal{F}, P)$$, @@ -91,7 +93,7 @@ $$X^{-1}$$ 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 often stated as "$$X$$ is *$$\mathcal{F}$$-measurable*". Related to $$\mathcal{F}$$ is the **information** obtained by observing a random variable $$X$$. @@ -111,7 +113,7 @@ 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"* +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 $$Y$$. @@ -145,11 +147,12 @@ $$\begin{aligned} \end{aligned}$$ + ## 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). +of every possible value of $$X$$ multiplied by the corresponding probability (density). For continuous and discrete sample spaces $$\Omega$$, respectively: $$\begin{aligned} @@ -163,7 +166,7 @@ $$\begin{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, +is the following *Lebesgue-Stieltjes integral*, since $$F_X(x)$$ always exists: $$\begin{aligned} @@ -172,7 +175,7 @@ $$\begin{aligned} \end{aligned}$$ This is valid for any sample space $$\Omega$$. -Or, equivalently, a Lebesgue integral can be used: +Or, equivalently, a *Lebesgue integral* can be used: $$\begin{aligned} \mathbf{E}[X] |