---- > [!definition] Definition. ([[probability density function]]) > Denote by $\lambda$ the [[Lebesgue measure]] on $\mathbb{R}$. Suppose $X$ is a [[random variable]] on some [[probability|probability space]]. If [[Lp-norm|there exists]] $h \in L^{1}(\mathbb{R})$ such [[distribution function|that]] $\widetilde{X}(s)=\int _{-\infty}^{s}h \, d \lambda $ for all $s \in \mathbb{R}$, then $h$ is called the **density function** of $X$. ^definition Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[probability space]] and $(E, \mathcal{E})$ a [[σ-algebra|measurable space]]. Given a reference [[measure]] $\mu$ on $(E, \mathcal{E})$ [[absolutely continuous|with]] $\mathbb{P}_{X} \ll \mu$, the [[Radon-Nikodym Theorem|Radon-Nikodym derivative]] $h=\frac{d \mathbb{P}_{X}}{d\mu} \in L^{1}(\mu)$ is called the **probability density function wrt $\mu$ (PDF)** of $X$. Usually $E=\mathbb{R}$ and $\mu=\lambda$ is the [[Lebesgue measure]] whence $h$ [[integral|integrates]] to the [[distribution function|CDF]]: $\widetilde{X}(s)=\int _{-\infty}^{s} h \, d \lambda .$ In what follows, $\mathcal{B}$ denotes the [[Borel set|Borel σ-algebra]] on $\mathbb{R}$. > [!definition] Definition. (Random variable generated by a PDF) > Suppose $h \in L^{1}(\mathbb{R})$ is such that $\int _{-\infty}^{\infty} h \, d \lambda=1$ and $h(x) \geq 0$ for [[almost-everywhere|almost every]] $x \in \mathbb{R}$. Let $\mathbb{P}$ be the [[measure with a density|measure with density]] $h$ wrt $\lambda$, that is, let $\mathbb{P}$ be the [[probability|probability measure]] on $(\mathbb{R}, \mathcal{B})$ given by $\mathbb{P}(B):= \int_{B} h \, d \lambda.$ Then $h$ is the density function of the [[random variable]] $X$ on $(\mathbb{R}, \mathcal{B})$ defined by $X(x):=x$ for $x \in \mathbb{R}$.[^1] We call $X$ the **random variable generated by** $h$. ^definition > [!basicproperties] Basic properties of the random variable generated by a PDF. > - *([[expectation]])* $\mathbb{E}X=\int _{-\infty}^{\infty} xh(x) \, d \lambda(x)$ > - *([[variance]])* If $X \in \mathcal{L}^{2}(\mathbb{P})$, then $\operatorname{var }X=\int _{-\infty}^{\infty } x^{2}h(x) \, d \lambda(x)- \big(\int _{-\infty}^{\infty} xh(x) \, d \lambda(x) \big)^{2}.$ > [^1]: Indeed, $\widetilde{X}(s)=\mathbb{P}(X \leq s)=\int _{\{ x \in \mathbb{R}: x \leq s \}} h\, d \lambda$. ---- #### ---- #### 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 # A [[continuous]] Analogue to the [[discrete random variable|PMF]] (PMF) ![[CleanShot 2022-09-25 at 17.27.25 1.jpg]] # Obtaining Probabilities from Density and [[distribution function]] Take a [[Probability Density Function]] curve like this: ![[CleanShot 2022-10-20 at [email protected]]] Imagine a [[discrete]] analogue: ![[CleanShot 2022-10-20 at [email protected]]] How would you find $P(X < 5)$ for the above figure? You would consider it a union of disjoint events (so like [[the exclusion rule]] but intersections all empty), $P(X<5) = P(X=1) + P(X=2) + P(X=3) + P(X=4) = \sum_{i=1}^4 X_i.$ In order to generalize to the [[continuous]] case, we take an [[integral]] instead of a sum: $P(X<5) = \int_{-\infty}^5 \text{(our ftn)}.$ #notFormatted