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The expected value of any real-valued random variable $X$ can also be defined on the graph of its cumulative distribution function $F$ by a nearby equality of areas. In fact, $\operatorname {E} [X]=\mu$ with a real number $\mu$ if and only if the two surfaces in the $x$ - $y$ -plane, described by
$\quad \,x\leq \mu ,\;\,0\leq y\leq F(x)\quad ^{\vphantom {\big |}}$ or $\quad \,x\geq \mu ,\;\,F(x)\leq y\leq 1_{\vphantom {\big |}}$
respectively, have the same finite area, i.e. if $\int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx$ and both improper Riemann integrals converge. Finally, this is equivalent to the representation $\operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,$ also with convergent integrals.^[1]

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The expected value of any real-valued random variable $X$ can also be defined on the graph of its cumulative distribution function $F$ by a nearby equality of areas. In fact, $\operatorname {E} [X]=\mu$ with a real number $\mu$ if and only if the two surfaces in the $x$ - $y$ -plane, described by

x\leq \mu ,\;\,0\leq y\leq F(x)\quad

or

\quad x\geq \mu ,\;\,F(x)\leq y\leq 1

respectively, have the same finite area, i.e. if $\int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx$ and both improper Riemann integrals converge. Finally, this is equivalent to the representation $\operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,$ also with convergent integrals.^[2]

XXXXXXXXXXXXX

The expected value of any real-valued random variable $X$ can also be defined on the graph of its cumulative distribution function $F$ by a nearby equality of areas. In fact, $\operatorname {E} [X]=\mu$ with a real number $\mu$ if and only if the two surfaces in the $x$ - $y$ -plane, described by $x\leq \mu ,\;\,0\leq y\leq F(x)\quad {\text{or}}\quad x\geq \mu ,\;\,F(x)\leq y\leq 1$ respectively, have the same finite area, i.e. if $\int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx$ and both improper Riemann integrals converge. Finally, this is equivalent to the representation $\operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,$ also with convergent integrals.^[3]

^ Uhl, Roland (2023). Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion (PDF). Technische Hochschule Brandenburg. doi:10.25933/opus4-2986. pp. 2–4.
^ Uhl, Roland (2023). Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion (PDF). Technische Hochschule Brandenburg. doi:10.25933/opus4-2986. pp. 2–4.
^ Uhl, Roland (2023). Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion (PDF). Technische Hochschule Brandenburg. doi:10.25933/opus4-2986. pp. 2–4.

[1] Uhl, Roland (2023). Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion (PDF). Technische Hochschule Brandenburg. doi:10.25933/opus4-2986. pp. 2–4.

[2] Uhl, Roland (2023). Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion (PDF). Technische Hochschule Brandenburg. doi:10.25933/opus4-2986. pp. 2–4.

[3] Uhl, Roland (2023). Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion (PDF). Technische Hochschule Brandenburg. doi:10.25933/opus4-2986. pp. 2–4.

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