---- > [!definition] Definition. ([[distribution function]] - general analysis) > Suppose $(X, \Sigma, \mu)$ is a [[measure|measure space]] and $f:X \to \mathbb{R}$ is a [[measurable function]]. The **distribution function** or **survival function** of $f$ is the function $d_{f}:\mathbb{R} \to \mathbb{R}$ given by $d_{f}(s):= \mu (f > s).$ ^definition > [!definition] Definition. ([[distribution function|(cumulative) distribution function]] - probability) > Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a [[probability|probability space]] and $X:\Omega \to \mathbb{R}$ is a [[random variable]]. The **(cumulative) distribution function (CDF)** of $X$ is defined $F_{X}:=1-d_{X}$. (It is also denoted $\widetilde{X}$.) Explicitly, $F_{X}$ is the function $\begin{align} \mathbb{R} &\to [0,1] \\ s & \mapsto \mathbb{P}_{X}\big( (-\infty, s) ] = \mathbb{P}(X \leq s), \end{align}$ where $\mathbb{P}_{X}$ denotes the [[probability distribution]] of $X$. > [!equivalence] > - [[characterization of cumulative distribution functions]] ^equivalence > [!basicexample] > Suppose $A \in \mathcal{F}$ is an [[probability|event]]. [[indicator random variable|Then]] $\widetilde{1_{A}}(s)=\begin{cases} 0 & s < 0 \\ 1-\mathbb{P}(A) & 0 \leq s < 1 \\ 1 & s \geq 1. \end{cases}$ ([[probability distribution|Recall]] $\mathbb{P}_{1_{A}}=(1-\mathbb{P}(A))\delta_{0} + \mathbb{P}(A) \delta_{1}$ .) ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #analysis/probability-statistics # Definition Let $\Omega$ be our [[sample space]]. Suppose we have a [[random variable]] $X: \Omega \to \mathbb{R}$. Then the [[distribution function]], denoted $F_X: \mathbb{R} \to [0,1]$ is defined as $F_X(a) := P(\omega:X(\omega) \leq a).$ 'probability of the [[event]] corresponding to set of all $\omega$ for which $X(\omega)$ does not exceed the input $a$ occuring' ### Coin-Tossing Example ![[CleanShot 2022-09-23 at 19.59.53.jpg]] # Properties of the [[distribution function]] *In general:* 1. $0 \leq F_X(a) \leq 1 \ \ \ \forall a \in \mathbb{R}$ ([[why?#34]]) 2. $a < b \implies F_X(a) \leq F_X(b)$ (non-decreasing) ([[why?#35]]) 3. $\lim_{a \to \infty} F_X(a) = 0$; $\lim_{a \to \infty} F_X(a) = 1$ 4. If $(a_n) \to a_+$, then $\lim_{n \to \infty}F_X(a_n) = F_X(a)$ ('right-hand semi-[[continuous]]') Note: ![[CleanShot 2022-09-23 at 20.15.28.jpg]] ![[CleanShot 2022-09-23 at 20.16.26.jpg]] # Important! If a function $F: \mathbb{R} \to [0,1]$ satisfies above properties 1-4, then $\exists$ a [[random variable]] $X: \Omega \to \mathbb{R}$ such that $F_X = F$. ## Proof — Not in 525 #notFormatted #analysis/probability-statistics # definition — distribution function the **distribution function** of a [[random variable]] $X$ is the function $F: \mathbb{R} \to [0,1]$ given by $F(x) = \mathbb{P}(X \leq x)$. Often we denote it by $F_{X}$. ### Equivalent Lemma A [[distribution]] function $F$ has the following properties: 1. $\lim_{ x \to -\infty } F(x)=0, \lim_{ x \to \infty } F(x)=1$ 2. if $x<y$, then $F(x) \leq F(y)$ 3. $F$ is right-[[continuous]], that is, $F(x+h) \to F(x) as $h \downarrow 0$. #### $F$ is a distribution function IFF it satsifes all 3 of the above conditions! ## Another Lemma Let $F$ be a distribution function of $X$. Then 1. $\mathbb{P}(X>x) = 1-F(x)$ 2. $\mathbb{P}(x < X \leq y) = F(y)-F(x)$ 3. $\mathbb{P}(X=x) = F(x) - \lim_{y \uparrow x} F(y)$ #notFormatted