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rename "introducci\303\263n_estad\303\255stica_multivariante_AD.Rmd" => "teoria/Multivariante/introducci\303\263n_estad\303\255stica_multivariante_AD.Rmd" (100%) rename "introducci\303\263n_estad\303\255stica_multivariante_AD.log" => "teoria/Multivariante/introducci\303\263n_estad\303\255stica_multivariante_AD.log" (100%) rename "introducci\303\263n_estad\303\255stica_multivariante_AD.pdf" => "teoria/Multivariante/introducci\303\263n_estad\303\255stica_multivariante_AD.pdf" (100%) diff --git a/ejercicios/Propuestos/Tema-4---Momentos_1.Rmd b/ejercicios/Propuestos/Tema-4---Momentos_1.Rmd new file mode 100755 index 0000000..d159d77 --- /dev/null +++ b/ejercicios/Propuestos/Tema-4---Momentos_1.Rmd @@ -0,0 +1,76 @@ +--- +title: "Ejercicios Tema 4 - Complementos de Variables aleatorias" +author: "Ricardo Alberich, Juan Gabriel Gomila y Arnau Mir" +date: "Curso de Probabilidad y Variables Aleatorias con R y Python" +output: + pdf_document: default + html_document: + df_print: paged +header-includes: \usepackage{amsmath} +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE) +``` + +\newcommand\momento{m} +\newcommand{\momentocentral}{\mu} +\newcommand{\FunGenMom}{m} +\newcommand{\FunCar}{\phi} + +# Momentos y momentos centrados. Asimetría y apuntamiento. + +## Pregunta 1. + + Halla el momento de orden $n$ y el momento centrado de orden $n$ para la variable aleatoria $W$ con función de distribución: +$$ +F_W (t)= +\left\{\begin{array}{ll} +0, & \mbox{si $t<3$}, + \\[0.1cm] +\frac13, & \mbox{si $3\leq t<4$}, + \\[0.1cm] +\frac12, & \mbox{si $4\leq t<5$}, + \\[0.1cm] +\frac23, & \mbox{si $5\leq t<6$}, + \\[0.1cm] +1, & \mbox{si $t\geq 6$}, +\end{array}\right. +$$ + + +## Pregunta 2. + +Halla el momento de orden $n$ y el momento central de orden $n$ para la variable aleatoria $Z$ con función de probabilidad: +$$f_Z (x)= +\left\{\begin{array}{ll} +{1\over 3}, & \mbox{si $x=0,1,2$},\\ 0, & \mbox{en los otros +casos.} +\end{array}\right. +$$ + + +## Pregunta 3. + +Halla el momento de orden $n$ y el momento central de orden $n$ para la variable aleatoria $X$ con función de densidad: + +$$ +f(x)=\begin{cases} +1-|x|, & \mbox{si }|x|\leq 1,\\ +0, & \mbox{en caso contrario.} +\end{cases} +$$ + +## Pregunta 4. + +Halla el momento de orden $n$ y el momento central de orden $n$ para la variable aleatoria $Y$ con función de distribución: + +$$ +F_Y(t)= +\left\{\begin{array}{ll} +0, & \mbox{si $t<0$},\\ +\sqrt{t}, & \mbox{si $0\leq t\leq 1$},\\ 1, & +\mbox{si $t>1$}, +\end{array}\right. +$$ + diff --git a/ejercicios/Propuestos/Tema-4---Momentos_1.log b/ejercicios/Propuestos/Tema-4---Momentos_1.log old mode 100644 new mode 100755 index 1563a38..7207c4d --- a/ejercicios/Propuestos/Tema-4---Momentos_1.log +++ b/ejercicios/Propuestos/Tema-4---Momentos_1.log @@ -1,98 +1,110 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/W32TeX) (preloaded format=pdflatex 2022.3.24) 31 MAR 2022 18:48 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023) (preloaded format=pdflatex 2024.2.26) 28 FEB 2024 16:36 entering extended mode restricted \write18 enabled. %&-line parsing enabled. -**Tema-4---Momentos_1.tex -(./Tema-4---Momentos_1.tex -LaTeX2e <2021-11-15> patch level 1 -L3 programming layer <2022-02-24> (c:/Users/t169/AppData/Roaming/TinyTeX/texmf- -dist/tex/latex/base/article.cls -Document Class: article 2021/10/04 v1.4n Standard LaTeX document class -(c:/Users/t169/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/base/size10.clo -File: size10.clo 2021/10/04 v1.4n Standard LaTeX file (size option) -) -\c@part=\count185 -\c@section=\count186 -\c@subsection=\count187 -\c@subsubsection=\count188 -\c@paragraph=\count189 -\c@subparagraph=\count190 -\c@figure=\count191 -\c@table=\count192 -\abovecaptionskip=\skip47 -\belowcaptionskip=\skip48 -\bibindent=\dimen138 -) -(c:/Users/t169/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/amsmath/amsmath.sty -Package: amsmath 2021/10/15 v2.17l AMS math features -\@mathmargin=\skip49 +**Tema--4---Momentos_Enunciados_1.tex +(./Tema--4---Momentos_Enunciados_1.tex +LaTeX2e <2023-11-01> patch level 1 +L3 programming layer <2024-02-20> +(c:/Users/ricuib/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/base/article.cls +Document Class: article 2023/05/17 v1.4n Standard LaTeX document class +(c:/Users/ricuib/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/base/size10.clo +File: size10.clo 2023/05/17 v1.4n Standard LaTeX file (size option) +) +\c@part=\count188 +\c@section=\count189 +\c@subsection=\count190 +\c@subsubsection=\count191 +\c@paragraph=\count192 +\c@subparagraph=\count193 +\c@figure=\count194 +\c@table=\count195 +\abovecaptionskip=\skip48 +\belowcaptionskip=\skip49 +\bibindent=\dimen140 +) (c:/Users/ricuib/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2023/05/13 v2.17o AMS math features +\@mathmargin=\skip50 For additional information on amsmath, use the `?' option. - 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Asimetría y apuntamiento. + +## Pregunta 1. + + Halla el momento de orden $n$ y el momento centrado de orden $n$ para la variable aleatoria $W$ con función de distribución: +$$ +F_W (t)= +\left\{\begin{array}{ll} +0, & \mbox{si $t<3$}, + \\[0.1cm] +\frac13, & \mbox{si $3\leq t<4$}, + \\[0.1cm] +\frac12, & \mbox{si $4\leq t<5$}, + \\[0.1cm] +\frac23, & \mbox{si $5\leq t<6$}, + \\[0.1cm] +1, & \mbox{si $t\geq 6$}, +\end{array}\right. +$$ + +### Solución + +No es complicado, revisa las clases de teoría si tienes algún problema con alguna definición. + +Como es discreta tenemos que + +$$P_W (t)= +\left\{\begin{array}{ll} +\frac26, & \mbox{si $t=3$}, + \\ [0.1cm] +\frac16, & \mbox{si $t=4$}, + \\ [0.1cm] +\frac16, & \mbox{si $t=5$}, + \\ [0.1cm] +\frac26, & \mbox{$t=6$},\\ +0, & \mbox{en cualquier otro caso}. +\end{array}\right. +$$ + + +El momento respecto al origen de orden $n$ es: + +$$ +\begin{aligned} +E(W^n)&=\sum_{t=3}^6 t\cdot P(W=t)= +3^n\cdot \frac26+ +4^n\cdot \frac16+ +5^n\cdot \frac16+ +6^n\cdot \frac26 \\ +&=\frac{3^n\cdot 2+4^n\cdot 1+5^n\cdot 1+6^n\cdot 2}{6}= +\frac{3^n\cdot 2+4^n+5^n+6^n \cdot 2}{6}. +\end{aligned} +$$ + +Para $n=1$ es $E(W)=\frac{27}{6}=\frac{9}{2}=4.5,$ y por lo tanto en momento centrado de orden $n$ es + + +$$ +\begin{aligned} +E((W-E(W))^n) &= E((W-4.5)^n) =\sum_{t=3}^6 \left(t-4.5\right)^n\cdot P(W=t) \\ +&= +\left(3-4.5\right)^n \cdot \frac26+ +\left(4-4.5\right)^n \cdot \frac16+ +\left(5-4.5\right)^n \cdot \frac16+ +\left(6-4.5\right)^n \cdot \frac26 \\ +&= \frac{(-1.5)^n\cdot 2+(-1)^n\cdot 0.5^n+1\cdot 0.5^n+ 2\cdot 1.5^n}{6}\\ +&= \frac{2\cdot (-1^n)\cdot \left(\frac32\right)^n+(-1)^n\cdot \left(\frac12\right)^n+ \left(\frac12\right)^n+ +2\cdot \left(\frac32\right)^n}{6}. +\end{aligned} +$$ +Finalmente + +$$ +E((W-E(W))^n)=\left\{\begin{array}{ll} +\frac{-2\cdot \left(\frac32\right)^n-\left(\frac12\right)^n+ \left(\frac12\right)^n+ +2\cdot \left(\frac32\right)^n}{6}=0 & \mbox{si } n=1,3,5,7...\mbox{ impar,} \\ +\frac{2\cdot \left(\frac32\right)^n+ \left(\frac12\right)^n+ \left(\frac12\right)^n+ +2\cdot \left(\frac32\right)^n}{6}=\frac{4\cdot \left(\frac32\right)+2\left(\frac12\right) }{6}= +\frac{2\cdot \left(\frac32\right)^n+\left(\frac12\right)^n }{3}& \mbox{si } n=2,4,6...\mbox{ par}. +\end{array} +\right. +$$ + + + + + +## Pregunta 2. + +Halla el momento de orden $n$ y el momento central de orden $n$ para la variable aleatoria $Z$ con función de probabilidad: +$$f_Z (x)= +\left\{\begin{array}{ll} +{1\over 3}, & \mbox{si $x=0,1,2$},\\ 0, & \mbox{en los otros +casos.} +\end{array}\right. +$$ + +### Solución + +El momento respecto al origen de orden $n$ es: + +$$ +E(Z^n) =\sum_{x=0}^2 x^n\cdot P(Z=x)=0^n\cdot \frac13+ 1^n\cdot \frac13+2^n \cdot \frac13=\frac{1+2^n}{3}. +$$ + +La esperanza es del momento respecto al origen para $n=1$ y es $E(Z)=E(Z^1)=\frac{1+2^1}{3}=1.$ + +Así el momento central de orden $n$ es + +$$ +\begin{aligned} +E((Z-E(Z))^n)& = E((Z-1)^n)=\sum_{x=0}^2 (x-1)^n\cdot P(Z=x)\\ +&= (0-1)^n\cdot \frac13+ (1-1)^n\cdot \frac13+(2-1)^n \cdot \frac13\\ +&=\frac{(-1)^n+0^n+1^n}{3}=\frac{(-1)^n+1}{3}. +\end{aligned} +$$ + +Luego + +$$ +E((Z-E(Z))^n)= E((Z-1)^n)= +\left\{ +\begin{array}{ll} +\frac{(-1)^n+1}{3} =\frac{-1+1}{3}=0, & \mbox{si } n=1,3,5,7...\mbox{ es impar} \\ +\frac{(-1)^n+1}{3} =\frac{1+1}{3}=\frac{2}{3}. & \mbox{si } n=2,4,6...\mbox{par}. +\end{array} +\right. +$$ + +## Pregunta 3. + +Halla el momento de orden $n$ y el momento central de orden $n$ para la variable aleatoria $X$ con función de densidad: + +$$ +f(x)=\begin{cases} +1-|x|, & \mbox{si }|x|\leq 1,\\ +0, & \mbox{en caso contrario.} +\end{cases} +$$ + +### Solución + +Notemos que + +$$ +f(x)=\begin{cases} +1+x, & \mbox{si } -1 \leq x \leq 0,\\ +1-x, & \mbox{si } 0 \leq x \leq 1,\\ +0, & \mbox{en caso contrario.} +\end{cases} +$$ + +En este caso el momento de orden $n$ respecto al origen es + + +$$ +\begin{aligned} +E(X^n)&= \int_{-1}^1 x^n \cdot f(x) dx= \int_{-1}^0 x^n \cdot (1+x) dx+ +\int_{0}^1 x^n \cdot (1-x) dx\\ +& = +\left[\frac{x^{n+1}}{n+1}+\frac{x^{n+2}}{n+2}\right]_{x=-1}^0+ +\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_{x=0}^1\\ +& = +0-\left(\frac{(-1)^{n+1}}{n+1}+\frac{(-1)^{n+2}}{n+2}\right)+ +\left(\frac{1^{n+1}}{n+1}-\frac{1^{n+2}}{n+2}\right)-0\\ & = +\left(\frac{(-1)^{n+2}}{n+1}+\frac{(-1)^{n+3}}{n+2}\right)+ +\left(\frac{1}{n+1}-\frac{1}{n+2}\right)\\ & = +(-1)^{n+2}\cdot \left(\frac{1}{n+1}-\frac{1}{n+2}\right)+ +\left(\frac{1}{n+1}-\frac{1}{n+2}\right)= +\\ & = +(-1)^{n+2}\cdot \left(\frac{(n+2)-(n+1)}{(n+1)\cdot (n+2)}\right)+ +\left(\frac{n+2-(n+1)}{(n+1)\cdot (n+2)}\right)\\ +& = +\left((-1)^{n+2}+1\right)\cdot\left(\frac{1}{(n+1)\cdot(n+2)}\right) \\ & = +\left\{ +\begin{array}{rl} +0, & \mbox{si } n=1,3,5,7... \mbox{ es impar} \\ +\frac{2}{(n+1)\cdot(n+2)}, & \mbox{si } n=2,4,6...\mbox{ es par}. +\end{array} +\right. +\end{aligned} +$$ +Así que la esperanza vale $E(X)=0$. + + +Luego el momento respecto a la media de orden $n$ es igual al momento del mismo orden respecto al origen: + +$$E((X-E(X))^n)=E(X^n)= +\left\{ +\begin{array}{ll} +0, & \mbox{si } n=1,3,5,7... \mbox{ es impar} \\ +\frac{2}{(n+1)\cdot(n+2)}. & \mbox{si } n=2,4,6...\mbox{ es par}. +\end{array} +\right. +$$ + + +## Pregunta 4. + +Halla el momento de orden $n$ y el momento central de orden $n$ para la variable aleatoria $Y$ con función de distribución: + +$$ +F_Y(t)= +\left\{\begin{array}{ll} +0, & \mbox{si $t<0$},\\ +\sqrt{t}, & \mbox{si $0\leq t\leq 1$},\\ 1, & +\mbox{si $t>1$}, +\end{array}\right. +$$ + +### Solución + +Ante todo la densidad es $f_y(t)== + +$$ +\left\{\begin{array}{ll} +0, & \mbox{si $t<0$},\\ +\frac{1}{2} \cdot {t}^{-\frac12}, +& \mbox{si $0\leq t\leq 1$},\\ 0, & +\mbox{en otro caso}, +\end{array}\right. +$$ + + +$$ +\begin{aligned} +E(X^n)&= \int_0^1 t^n \cdot \frac{1}{2} t^{\frac{-1}{2}} \cdot dt= +\frac12\cdot \int_0^1 t^{n-\frac{1}{2}} \cdot dt= +\frac12\cdot \left[\frac{t^{n-\frac{1}{2}+1}}{n-\frac{1}{2}+1}\right]_{t=0}^1\\ & = +\frac12\cdot \left[\frac{t^{n+\frac12}}{n+\frac12}\right]_{t=0}^1= \frac{1}{2\cdot n+1}. +\end{aligned} +$$ + + + +El valor esperado es $E(X)=\frac{1}{1+\frac12}=\frac13.$ + +Así que el momento central de orden $n$ es + +$$ +\begin{aligned} +E\left(\left(X-\frac13\right)^n\right) &= +\int_0^1 \left(t-\frac13\right)^n \cdot \frac12 \cdot t^{-\frac12}\cdot dt = +\frac12\cdot \int_0^1 \sum_{k=0}^n \binom{n}{k} \cdot t^{n-k}\cdot +\left(-\frac13\right)^k \cdot t^{-\frac12} dt \\ & = +\frac12\cdot \sum_{k=0}^n \binom{n}{k}\cdot \left(-\frac{1}{3}\right)^k\cdot \int_0^1 t^{n-k-\frac12}\cdot dt \\ & = +\frac12\cdot\sum_{k=0}^n \binom{n}{k} \cdot \left(-\frac{1}{3}\right)^k\cdot +\left[\frac{t^{n-k-\frac12+1}}{n-k-\frac12+1} \right]_{t=0}^1 \\ & = +\frac12\cdot\sum_{k=0}^n \binom{n}{k}\cdot \left(-\frac{1}{3}\right)^k\cdot +\frac{1^{n-k+\frac32}}{n-k+\frac12}-0\\ & = +\frac12\cdot\sum_{k=0}^n \binom{n}{k}\cdot \left(-\frac{1}{3}\right)^k\cdot +\frac{1}{n-k+\frac12}. +\end{aligned} +$$ + + + diff --git a/ejercicios/Resueltos/Tema--4---Momentos_Soluciones_1.aux b/ejercicios/Resueltos/Tema--4---Momentos_Soluciones_1.aux new file mode 100755 index 0000000..dbd75cf --- /dev/null +++ b/ejercicios/Resueltos/Tema--4---Momentos_Soluciones_1.aux @@ -0,0 +1,33 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\providecommand\BKM@entry[2]{} 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+ + + + + + + +
+

Momentos y momentos centrados. Asimetría y apuntamiento.

+
+

Pregunta 1.

+

Halla el momento de orden \(n\) y el +momento centrado de orden \(n\) para la +variable aleatoria \(W\) con función de +distribución: \[ +F_W (t)= +\left\{\begin{array}{ll} +0, & \mbox{si $t<3$}, +\\[0.1cm] +\frac13, & \mbox{si $3\leq t<4$}, +\\[0.1cm] +\frac12, & \mbox{si $4\leq t<5$}, +\\[0.1cm] +\frac23, & \mbox{si $5\leq t<6$}, +\\[0.1cm] +1, & \mbox{si $t\geq 6$}, +\end{array}\right. +\]

+
+

Solución

+

No es complicado, revisa las clases de teoría si tienes algún +problema con alguna definición.

+

Como es discreta tenemos que

+

\[P_W (t)= +\left\{\begin{array}{ll} +\frac26, & \mbox{si $t=3$}, +\\ [0.1cm] +\frac16, & \mbox{si $t=4$}, +\\ [0.1cm] +\frac16, & \mbox{si $t=5$}, +\\ [0.1cm] +\frac26, & \mbox{$t=6$},\\ +0, & \mbox{en cualquier otro caso}. +\end{array}\right. +\]

+

El momento respecto al origen de orden \(n\) es:

+

\[ +\begin{aligned} +E(W^n)&=\sum_{t=3}^6 t\cdot P(W=t)= +3^n\cdot \frac26+ +4^n\cdot \frac16+ +5^n\cdot \frac16+ +6^n\cdot \frac26 \\ +&=\frac{3^n\cdot 2+4^n\cdot 1+5^n\cdot 1+6^n\cdot 2}{6}= +\frac{3^n\cdot 2+4^n+5^n+6^n \cdot 2}{6}. +\end{aligned} +\]

+

Para \(n=1\) es \(E(W)=\frac{27}{6}=\frac{9}{2}=4.5,\) y por +lo tanto en momento centrado de orden \(n\) es

+

\[ +\begin{aligned} +E((W-E(W))^n) &= E((W-4.5)^n) =\sum_{t=3}^6 +\left(t-4.5\right)^n\cdot P(W=t) \\ +&= +\left(3-4.5\right)^n \cdot \frac26+ +\left(4-4.5\right)^n \cdot \frac16+ +\left(5-4.5\right)^n \cdot \frac16+ +\left(6-4.5\right)^n \cdot \frac26 \\ +&= \frac{(-1.5)^n\cdot 2+(-1)^n\cdot 0.5^n+1\cdot 0.5^n+ 2\cdot +1.5^n}{6}\\ +&= \frac{2\cdot (-1^n)\cdot \left(\frac32\right)^n+(-1)^n\cdot +\left(\frac12\right)^n+ \left(\frac12\right)^n+ +2\cdot \left(\frac32\right)^n}{6}. +\end{aligned} +\] Finalmente

+

\[ +E((W-E(W))^n)=\left\{\begin{array}{ll} +\frac{-2\cdot \left(\frac32\right)^n-\left(\frac12\right)^n+ +\left(\frac12\right)^n+ +2\cdot \left(\frac32\right)^n}{6}=0 & \mbox{si } n=1,3,5,7...\mbox{ +impar,} \\ +\frac{2\cdot \left(\frac32\right)^n+ \left(\frac12\right)^n+ +\left(\frac12\right)^n+ +2\cdot \left(\frac32\right)^n}{6}=\frac{4\cdot +\left(\frac32\right)+2\left(\frac12\right) }{6}= +\frac{2\cdot \left(\frac32\right)^n+\left(\frac12\right)^n }{3}& +\mbox{si } n=2,4,6...\mbox{ par}. +\end{array} +\right. +\]

+
+
+
+

Pregunta 2.

+

Halla el momento de orden \(n\) y el +momento central de orden \(n\) para la +variable aleatoria \(Z\) con función de +probabilidad: \[f_Z (x)= +\left\{\begin{array}{ll} +{1\over 3}, & \mbox{si $x=0,1,2$},\\ 0, & \mbox{en los otros +casos.} +\end{array}\right. +\]

+
+

Solución

+

El momento respecto al origen de orden \(n\) es:

+

$$ E(Z^n) =_{x=0}^2 xnP(Z=x)=0n+ 1n+2n =.

+

$$

+

La esperanza es del momento respecto al origen para \(n=1\) y es \(E(Z)=E(Z^1)=\frac{1+2^1}{3}=1.\)

+

Así el momento central de orden \(n\) es

+

\[ +\begin{aligned} +E((Z-E(Z))^n)& = E((Z-1)^n)=\sum_{x=0}^2 (x-1)^n\cdot P(Z=x)\\ +&= (0-1)^n\cdot \frac13+ (1-1)^n\cdot \frac13+(2-1)^n \cdot +\frac13\\ +&=\frac{(-1)^n+0^n+1^n}{3}=\frac{(-1)^n+1}{3}. +\end{aligned} +\]

+

Luego

+

\[ +E((Z-E(Z))^n)= E((Z-1)^n)= +\left\{ +\begin{array}{ll} +\frac{(-1)^n+1}{3} =\frac{-1+1}{3}=0, & \mbox{si } +n=1,3,5,7...\mbox{ es impar} \\ +\frac{(-1)^n+1}{3} =\frac{1+1}{3}=\frac{2}{3}. & \mbox{si } +n=2,4,6...\mbox{par}. +\end{array} +\right. +\]

+
+
+
+

Pregunta 3.

+

Halla el momento de orden \(n\) y el +momento central de orden \(n\) para la +variable aleatoria \(X\) con función de +densidad:

+

\[ +f(x)=\begin{cases} +1-|x|, & \mbox{si }|x|\leq 1,\\ +0, & \mbox{en caso contrario.} +\end{cases} +\]

+
+

Solución

+

Notemos que

+

\[ +f(x)=\begin{cases} +1+x, & \mbox{si } -1 \leq x \leq 0,\\ +1-x, & \mbox{si } 0 \leq x \leq 1,\\ +0, & \mbox{en caso contrario.} +\end{cases} +\]

+

En este caso el momento de orden \(n\) respecto al origen es

+

\[ +\begin{aligned} +E(X^n)&= \int_{-1}^1 x^n \cdot f(x) dx= \int_{-1}^0 x^n \cdot +(1+x) dx+ +\int_{0}^1 x^n \cdot (1-x) dx\\ +& = +\left[\frac{x^{n+1}}{n+1}+\frac{x^{n+2}}{n+2}\right]_{x=-1}^0+ +\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_{x=0}^1\\ +& = +0-\left(\frac{(-1)^{n+1}}{n+1}+\frac{(-1)^{n+2}}{n+2}\right)+ +\left(\frac{1^{n+1}}{n+1}-\frac{1^{n+2}}{n+2}\right)-0\\ & = +\left(\frac{(-1)^{n+2}}{n+1}+\frac{(-1)^{n+3}}{n+2}\right)+ +\left(\frac{1}{n+1}-\frac{1}{n+2}\right)\\ & = +(-1)^{n+2}\cdot \left(\frac{1}{n+1}-\frac{1}{n+2}\right)+ +\left(\frac{1}{n+1}-\frac{1}{n+2}\right)= +\\ & = +(-1)^{n+2}\cdot \left(\frac{(n+2)-(n+1)}{(n+1)\cdot (n+2)}\right)+ +\left(\frac{n+2-(n+1)}{(n+1)\cdot (n+2)}\right)\\ +& = +\left((-1)^{n+2}+1\right)\cdot\left(\frac{1}{(n+1)\cdot(n+2)}\right) \\ +& = +\left\{ +\begin{array}{ll} +0, & \mbox{si } n=1,3,5,7... \mbox{ es impar} \\ +\frac{2}{(n+1)\cdot(n+2)}. & \mbox{si } n=2,4,6...\mbox{ es par}. +\end{array} +\right. +\end{aligned} +\] Así que la esperanza vale \(E(X)=0\).

+

Luego el momento respecto a la media de orden \(n\) es igual al momento del mismo orden +respecto al origen:

+

\[E((X-E(X))^n)=E(X^n)= +\left\{ +\begin{array}{ll} +0, & \mbox{si } n=1,3,5,7... \mbox{ es impar} \\ +\frac{2}{(n+1)\cdot(n+2)}. & \mbox{si } n=2,4,6...\mbox{ es par}. +\end{array} +\right. +\]

+
+
+
+

Pregunta 4.

+

Halla el momento de orden \(n\) y el +momento central de orden \(n\) para la +variable aleatoria \(Y\) con función de +distribución:

+

\[ +F_Y(t)= +\left\{\begin{array}{ll} +0, & \mbox{si $t<0$},\\ +\sqrt{t}, & \mbox{si $0\leq t\leq 1$},\\ 1, & +\mbox{si $t>1$}, +\end{array}\right. +\]

+
+

Solución

+

\[ +\begin{aligned} +E(X^n)&= \int_0^1 t^n \cdot \sqrt{t}\cdot dt=\int_0^1 +t^{n+\frac{1}{2}} \cdot dt +=\left[\frac{t^{n+\frac{1}{2}+1}}{n+\frac{1}{2}+1}\right]_{t=0}^1\\ +& = +\left[\frac{t^{n+\frac{3}{2}}}{n+\frac{3}{2}}\right]_{t=0}^1= +\frac{1}{n+\frac32}. +\end{aligned} +\]

+

El valor esperado es \(E(X)=\frac{1}{1+\frac32}=\frac25.\)

+

Así que el momento central de orden \(n\) es

+

\[ +\begin{aligned} +E\left(\left(X-\frac25\right)^n\right) &= +\int_0^1 \left(t-\frac25\right)^n \cdot \sqrt{t} \cdot dt = +\int_0^1 \sum_{k=0}^n \binom{n}{k} \cdot t^{n-k}\cdot +\left(-\frac{3}{5}\right)^k \cdot t^{\frac12} dt \\ & = +\sum_{k=0}^n \binom{n}{k}\cdot +\left(-\frac{3}{5}\right)^k\cdot \int_0^1 t^{n-k+\frac12}\cdot dt \\ +& = +\sum_{k=0}^n \binom{n}{k}\cdot \left(-\frac{3}{5}\right)^k\cdot +\left[\frac{t^{n-k+\frac12+1}}{n-k+\frac12+1} \right]_{t=0}^1 \\ & += +\sum_{k=0}^n \binom{n}{k}\cdot \left(-\frac{3}{5}\right)^k\cdot +\frac{1^{n-k+\frac32}}{n-k+\frac32}-0\\ & = +\sum_{k=0}^n \binom{n}{k}\cdot \left(-\frac{3}{5}\right)^k\cdot +\frac{1}{n-k+\frac32} +\end{aligned} +\]

+
+
+
+ + + + +
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