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Merge pull request #100 from JoramSoch/master
added 2 definitions and 4 proofs
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‎D/max.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-12 05:33:00
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title: "Maximum"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further summary statistics"
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definition: "Maximum"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Sample maximum and minimum"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-12"
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url: "https://en.wikipedia.org/wiki/Sample_maximum_and_minimum"
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def_id: "D108"
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shortcut: "max"
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username: "JoramSoch"
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---
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**Definition:** The maximum of a sample or random variable is its highest observed or possible value.
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<br>
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1) Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a [sample](/D/samp) from a [random variable](/D/rvar) $X$. Then, the maximum of $x$ is
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$$ \label{eq:max-samp}
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\mathrm{max}(x) = x_j, \quad \text{such that} \quad x_j \geq x_i \quad \text{for all} \quad i = 1, \ldots, n, \; i \neq j \; ,
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$$
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i.e. the maximum is the value which is larger than or equal to all other observed values.
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<br>
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2) Let $X$ be a [random variable](/D/rvar) with possible values $\mathcal{X}$. Then, the maximum of $X$ is
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$$ \label{eq:max-rvar}
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\mathrm{max}(X) = \tilde{x}, \quad \text{such that} \quad \tilde{x} > x \quad \text{for all} \quad x \in \mathcal{X}\setminus\left\lbrace \tilde{x} \right\rbrace \; ,
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$$
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i.e. the maximum is the value which is larger than all other possible values.

‎D/min.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-12 05:25:00
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title: "Minimum"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further summary statistics"
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definition: "Minimum"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Sample maximum and minimum"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-12"
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url: "https://en.wikipedia.org/wiki/Sample_maximum_and_minimum"
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def_id: "D107"
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shortcut: "min"
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username: "JoramSoch"
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---
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**Definition:** The minimum of a sample or random variable is its lowest observed or possible value.
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<br>
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1) Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a [sample](/D/samp) from a [random variable](/D/rvar) $X$. Then, the minimum of $x$ is
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$$ \label{eq:min-samp}
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\mathrm{min}(x) = x_j, \quad \text{such that} \quad x_j \leq x_i \quad \text{for all} \quad i = 1, \ldots, n, \; i \neq j \; ,
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$$
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i.e. the minimum is the value which is smaller than or equal to all other observed values.
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<br>
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2) Let $X$ be a [random variable](/D/rvar) with possible values $\mathcal{X}$. Then, the minimum of $X$ is
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$$ \label{eq:min-rvar}
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\mathrm{min}(X) = \tilde{x}, \quad \text{such that} \quad \tilde{x} < x \quad \text{for all} \quad x \in \mathcal{X}\setminus\left\lbrace \tilde{x} \right\rbrace \; ,
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$$
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i.e. the minimum is the value which is smaller than all other possible values.

‎I/Table_of_Contents.md‎

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&emsp;&ensp; 1.4.4. *[Probability density function](/D/pdf)* <br>
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&emsp;&ensp; 1.4.5. **[Probability density function of strictly increasing function](/P/pdf-sifct)** <br>
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&emsp;&ensp; 1.4.6. **[Probability density function of strictly decreasing function](/P/pdf-sdfct)** <br>
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&emsp;&ensp; 1.4.7. *[Cumulative distribution function](/D/cdf)* <br>
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&emsp;&ensp; 1.4.8. **[Cumulative distribution function of strictly increasing function](/P/cdf-sifct)** <br>
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&emsp;&ensp; 1.4.9. **[Cumulative distribution function of strictly decreasing function](/P/cdf-sdfct)** <br>
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&emsp;&ensp; 1.4.10. *[Quantile function](/D/qf)* <br>
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&emsp;&ensp; 1.4.7. **[Probability density function in terms of cumulative distribution function](/P/pdf-cdf)** <br>
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&emsp;&ensp; 1.4.8. *[Cumulative distribution function](/D/cdf)* <br>
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&emsp;&ensp; 1.4.9. **[Cumulative distribution function of strictly increasing function](/P/cdf-sifct)** <br>
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&emsp;&ensp; 1.4.10. **[Cumulative distribution function of strictly decreasing function](/P/cdf-sdfct)** <br>
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&emsp;&ensp; 1.4.11. **[Cumulative distribution function of discrete random variable](/P/cdf-pmf)** <br>
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&emsp;&ensp; 1.4.12. **[Cumulative distribution function of continuous random variable](/P/cdf-pdf)** <br>
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&emsp;&ensp; 1.4.13. *[Quantile function](/D/qf)* <br>
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&emsp;&ensp; 1.4.14. **[Quantile function in terms of cumulative distribution function](/P/qf-cdf)** <br>
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&emsp;&ensp; 1.4.11. *[Moment-generating function](/D/mgf)* <br>
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&emsp;&ensp; 1.4.12. **[Moment-generating function of linear transformation](/P/mgf-ltt)** <br>
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&emsp;&ensp; 1.4.13. **[Moment-generating function of linear combination](/P/mgf-lincomb)** <br>

‎P/cdf-pdf.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-12 06:33:00
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title: "Cumulative distribution function in terms of probability density function of a continuous random variable"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Probability functions"
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theorem: "Cumulative distribution function of continuous random variable"
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sources:
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proof_id: "P190"
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shortcut: "cdf-pdf"
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username: "JoramSoch"
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---
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**Theorem:** Let $X$ be a [continuous](/D/rvar-disc) [random variable](/D/rvar) with possible values $\mathcal{X}$ and [probability density function](/D/pdf) $f_X(x)$. Then, the [cumulative distribution function](/D/cdf) of $X$ is
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$$ \label{eq:cdf-pdf}
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F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; .
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$$
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**Proof:** The [cumulative distribution function](/D/cdf) of a [random variable](/D/rvar) $X$ is defined as the probability that $X$ is smaller than $x$:
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$$ \label{eq:cdf}
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F_X(x) = \mathrm{Pr}(X \leq x) \; .
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$$
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The [probability density function](/D/pdf) of a [continuous](/D/rvar-disc) [random variable](/D/rvar) $X$ can be used to calculate the probability that $X$ falls into a particular interval $A$:
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$$ \label{eq:pdf}
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\mathrm{Pr}(X \in A) = \int_{A} f_X(x) \, \mathrm{d}x \; .
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$$
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Taking these two definitions together, we have:
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$$ \label{eq:cdf-pdf-qed}
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\begin{split}
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F_X(x) &\overset{\eqref{eq:cdf}}{=} \mathrm{Pr}(X \in (-\infty, x]) \\
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&\overset{\eqref{eq:pdf}}{=} \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; .
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\end{split}
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$$

‎P/cdf-pmf.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-12 06:03:00
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title: "Cumulative distribution function in terms of probability mass function of a discrete random variable"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Probability functions"
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theorem: "Cumulative distribution function of discrete random variable"
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sources:
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proof_id: "P189"
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shortcut: "cdf-pmf"
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username: "JoramSoch"
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---
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**Theorem:** Let $X$ be a [discrete](/D/rvar-disc) [random variable](/D/rvar) with possible values $\mathcal{X}$ and [probability mass function](/D/pmf) $f_X(x)$. Then, the [cumulative distribution function](/D/cdf) of $X$ is
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$$ \label{eq:cdf-pmf}
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F_X(x) = \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; .
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$$
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**Proof:** The [cumulative distribution function](/D/cdf) of a [random variable](/D/rvar) $X$ is defined as the probability that $X$ is smaller than $X$:
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$$ \label{eq:cdf}
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F_X(x) = \mathrm{Pr}(X \leq x) \; .
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$$
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The [probability mass function](/D/pmf) of a [discrete](/D/rvar-disc) [random variable](/D/rvar) $X$ returns the probability that $X$ takes a particular value $x$:
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$$ \label{eq:pmf}
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f_X(x) = \mathrm{Pr}(X = x) \; .
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$$
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Taking these two definitions together, we have:
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$$ \label{eq:cdf-pmf-qed}
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\begin{split}
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F_X(x) &\overset{\eqref{eq:cdf}}{=} \sum_{\overset{t \in \mathcal{X}}{t \leq x}} \mathrm{Pr}(X = t) \\
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&\overset{\eqref{eq:pmf}}{=} \sum_{\overset{t \in \mathcal{X}}{t \leq x}} f_X(t) \; .
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\end{split}
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$$

‎P/pdf-cdf.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-12 07:19:00
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title: "Probability density function is first derivative of cumulative distribution function"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Probability functions"
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theorem: "Probability density function in terms of cumulative distribution function"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Fundamental theorem of calculus"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-12"
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url: "https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Formal_statements"
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proof_id: "P191"
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shortcut: "pdf-cdf"
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username: "JoramSoch"
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---
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**Theorem:** Let $X$ be a [continuous](/D/rvar-disc) [random variable](/D/rvar). Then, the [probability distribution function](/D/pdf) of $X$ is the first derivative of the [cumulative distribution function](/D/cdf) of $X$:
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$$ \label{eq:pdf-cdf}
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f_X(x) = \frac{\mathrm{d}F_X(x)}{\mathrm{d}x} \; .
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$$
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**Proof:** The [cumulative distribution function in terms of the probability density function of a continuous random variable](/P/cdf-pdf) is given by:
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$$ \label{eq:cdf-pdf}
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F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t, \; x \in \mathbb{R} \; .
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$$
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Taking the derivative with respect to $x$, we have:
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$$ \label{eq:ddx-cdf}
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\frac{\mathrm{d}F_X(x)}{\mathrm{d}x} = \frac{\mathrm{d}}{\mathrm{d}x} \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \; .
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$$
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The fundamental theorem of calculus states that, if $f(x)$ is a continuous real-valued function defined on the interval $[a,b]$, then it holds that
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$$ \label{eq:FToC-1st}
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F(x) = \int_{a}^{x} f(t) \, \mathrm{d}t \quad \Rightarrow \quad F'(x) = f(x) \quad \text{for all} \quad x \in (a,b) \; .
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$$
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Applying \eqref{eq:FToC-1st} to \eqref{eq:cdf-pdf}, it follows that
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$$ \label{eq:pdf-cdf-qed}
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F_X(x) = \int_{-\infty}^{x} f_X(t) \, \mathrm{d}t \quad \Rightarrow \quad \frac{\mathrm{d}F_X(x)}{\mathrm{d}x} = f_X(x) \quad \text{for all} \quad x \in \mathbb{R} \; .
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$$

‎P/qf-cdf.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-11-12 07:48:00
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title: "Quantile function is inverse of strictly monotonically increasing cumulative distribution function"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Probability functions"
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theorem: "Quantile function in terms of cumulative distribution function"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Quantile function"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-11-12"
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url: "https://en.wikipedia.org/wiki/Quantile_function#Definition"
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proof_id: "P192"
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shortcut: "qf-cdf"
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username: "JoramSoch"
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---
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**Theorem:** Let $X$ be a [continuous](/D/rvar-disc) [random variable](/D/rvar) with the [cumulative distribution function](/D/cdf) $F_X(x)$. If the cumulative distribution function is strictly monotonically increasing, then the [quantile function](/D/qf) is identical to the inverse of $F_X(x)$:
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$$ \label{eq:qf-cdf}
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Q_X(p) = F_X^{-1}(x) \; .
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$$
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**Proof:** The [quantile function](/D/qf) $Q_X(p)$ is defined as the function that, for a given quantile $p \in [0,1]$, returns the smallest $x$ for which $F_X(x) = p$:
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$$ \label{eq:qf}
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Q_X(p) = \min \left\lbrace x \in \mathbb{R} \, \vert \, F_X(x) = p \right\rbrace \; .
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$$
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If $F_X(x)$ is continuous and strictly monotonically increasing, then there is only $x$ for which $F_X(x) = p$ and $F_X(x)$ is an invertible function, such that
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$$ \label{eq:qf-cdf-qed}
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Q_X(p) = F_X^{-1}(x) \; .
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$$

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