{"id":1068,"date":"2016-11-30T14:04:37","date_gmt":"2016-11-30T19:04:37","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=1068"},"modified":"2026-04-17T08:14:12","modified_gmt":"2026-04-17T13:14:12","slug":"variables-aleatorias-continuas-tabla","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-recursos\/variables-aleatorias-continuas-tabla\/","title":{"rendered":"Variables Aleatorias Cont\u00ednuas - Tabla"},"content":{"rendered":"\n

Referencia<\/strong>: Le\u00f3n-Garc\u00eda 445 Important Continuous Random Variables p164<\/em><\/p>\n\n\n\n

\n\n\n\n
\n

uniforme<\/a><\/p>\n\n\n\n

exponencial<\/a><\/p>\n\n\n\n

Normal<\/a><\/p>\n\n\n\n

Gamma<\/a><\/p>\n\n\n\n

Erlang m-1<\/a><\/p>\n\n\n\n

Chi-cuadrado<\/a><\/p>\n\n\n\n

Lapacian<\/a><\/p>\n\n\n\n

Rayleigh<\/a><\/p>\n\n\n\n

Cauchy<\/a><\/p>\n\n\n\n

Pareto<\/a><\/p>\n\n\n\n

Beta<\/a><\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Uniforme<\/h2>\n\n\n S_X = [a,b] <\/span>\n\n\n f_X(x) = \\frac{1}{b-a} <\/span>\n\n\n a\\leq x \\leq b <\/span>\n\n\n E[X] = \\frac{a+b}{2} <\/span>\n\n\n VAR[X] = \\frac{(b-a)^2}{12} <\/span>\n\n\n \\Phi_X(\\omega) = \\frac{e^{j\\omega b}- e^{j\\omega a}}{j\\omega(b-a)} <\/span>\n\n\n\n
\n\n\n\n

Exponencial<\/h2>\n\n\n S_X = [0, \\infty) <\/span>\n\n\n f_X(x) = \\lambda e ^{-\\lambda c}<\/span>\n\n\n x \\geq 0 \\text{ , } \\lambda > 0<\/span>\n\n\n E[X] = \\frac{1}{\\lambda} <\/span>\n\n\n VAR[X] = \\frac{1}{\\lambda ^2}<\/span>\n\n\n \\Phi_X(\\omega) = \\frac{\\lambda}{\\lambda - j\\omega} <\/span>\n\n\n\n

Nota: La variable aleatoria exponencial es la \u00fanica variable aleatoria cont\u00ednua con propiedad \"sin memoria\" <\/em><\/p>\n\n\n\n

\n\n\n\n

Normal o Gausiana<\/h3>\n\n\n S_X = (-\\infty, +\\infty) <\/span>\n\n\n f_X(x) = \\frac{e^{-(x-m)^2 \/2 \\sigma ^2}}{\\sqrt{2 \\pi}\\sigma} <\/span>\n\n\n\n

- \u221e < x < + \u221e , \u03c3 >0<\/p>\n\n\n E[X] = m <\/span>\n\n\n VAR[X] = \\sigma ^2 <\/span>\n\n\n \\Phi_X(\\omega) = e^{jm\\omega - \\sigma ^2 \\omega ^2 \/2} <\/span>\n\n\n\n

Nota: En un amplio rango de condiciones, X puede ser usada para aproximar la suma de un gran n\u00famero de variables aleatorias independientes <\/em><\/p>\n\n\n\n

\n\n\n\n

Gamma<\/h3>\n\n\n S_X = (0, \\infty) <\/span>\n\n\n f_X(x) = \\frac{ \\lambda (\\lambda x)^{\\alpha -1} e^{-\\lambda x} } {\\Gamma(\\alpha)} <\/span>\n\n\n x > 0, \\alpha >0, \\lambda > 0 <\/span>\n\n\n\n

donde \\Gamma(z) <\/span> es la funci\u00f3n gamma:<\/p>\n\n\n \\Gamma(z) = \\int_{0}^{\\infty} x^{z-1} e^{-x} dx , z>0<\/span>\n\n\n\n \\Gamma \\left(\\frac{1}{2} \\right) = \\sqrt{\\pi}<\/span>\n\n\n \\Gamma (z+1) = z \\Gamma(z) , z>0 <\/span>\n\n\n\n \\Gamma (m+1) = m! , m>0 \\text{, y entero}<\/span>\n\n\n E[x] = \\frac{\\alpha}{\\lambda} <\/span>\n\n\n VAR[X] = \\frac{\\alpha}{\\lambda^2}<\/span>\n\n\n \\Phi_X(\\omega) =\\frac{1}{(1-j\\omega \/\\lambda)^{\\alpha}} <\/span>\n\n\n\n

Casos especiales de Gamma<\/em><\/p>\n\n\n\n

Erlang m-1<\/h3>\n\n\n \\alpha = m \\text{, entero positivo} <\/span>\n\n\n\n

f_X(x) = \\frac{ \\lambda e^{-\\lambda x }(\\lambda x)^{m - 2} } {(m-1)!}<\/span> x>0<\/span><\/p>\n\n\n \\Phi_X(\\omega) = \\left( \\frac{1}{1-j\\omega \/\\lambda} \\right) ^{m}<\/span>\n\n\n\n

Nota: Una variable aleatoria Erlang m-1 se obtiene al a\u00f1adir m variables aleatorias exponenciales independientes con par\u00e1metro \u03bb <\/em><\/p>\n\n\n\n

Chi-cuadrado<\/h3>\n\n\n\n

con k grados de libertad:<\/p>\n\n\n \\alpha = k\/2 \\text{, k entero positivo y } \\lambda =1\/2<\/span>\n\n\n f_X(x) = \\frac{ x^{(k - 2)\/2} e^{-x\/2} } {2^{k\/2} \\Gamma (k\/2)}<\/span>\n\n\n x>0 <\/span>\n\n\n \\Phi_X(\\omega) = \\left( \\frac{1}{1-2j\\omega} \\right) ^{k\/2}<\/span>\n\n\n\n

Nota: la suma de k variables aleatorias Gausianas, mutuamente independientes, con varianza unitaria, media cuadrada 0, es una variable aleatoria Chi-cuadrada con k grados de libertad. <\/em><\/p>\n\n\n\n

\n\n\n\n

Lapacian<\/h3>\n\n\n S_X = (-\\infty, +\\infty) <\/span>\n\n\n f_X(x) = \\frac{\\alpha}{2} e^{-\\alpha|x|} <\/span>\n\n\n\n

- \u221e < x < + \u221e , \u03b1 >0<\/p>\n\n\n E[x] =0 <\/span>\n\n\n VAR[X] = \\frac{2}{\\alpha^2}<\/span>\n\n\n \\Phi_X(\\omega) = \\frac{\\alpha^2}{\\omega ^2 + \\alpha ^2} <\/span>\n\n\n\n

\n\n\n\n

Rayleigh<\/h3>\n\n\n S_X = [0, \\infty) <\/span>\n\n\n f_X(x) = \\frac{x}{\\alpha^2} e^{-x^2\/2\\alpha^2} <\/span>\n\n\n x \\geq 0 , \\alpha>0<\/span>\n\n\n E[X] = \\alpha \\sqrt{\\pi\/2}<\/span>\n\n\n VAR[X] =(2- \\pi\/2) \\alpha^2 <\/span>\n\n\n\n
\n\n\n\n

Cauchy<\/h3>\n\n\n S_X = (-\\infty, +\\infty) <\/span>\n\n\n f_X(x) = \\frac{\\alpha \/ \\pi}{x ^2 + \\alpha ^2}<\/span>\n\n\n\n

- \u221e < x < + \u221e , \u03b1 >0<\/p>\n\n\n\n

NO existe la media o varianza<\/p>\n\n\n \\Phi_X(\\omega) = e^{-\\alpha|\\omega| }<\/span>\n\n\n\n

\n\n\n\n

Pareto<\/h3>\n\n\n S_X = [X_m, +\\infty) , X_m>0<\/span>\n\n\n f_X(x) = \\begin{cases} 0 && x< x_m\\\\ \\alpha \\frac{x_m^{\\alpha}}{x^{\\alpha+1}} && x \\geq x_m \\end{cases} <\/span>\n\n\n E[X] =\\frac{\\alpha x_m}{\\alpha - 1}, \\alpha >1 <\/span>\n\n\n VAR[X] =\\frac{\\alpha x_m^2}{(\\alpha - 2)(\\alpha-1)^2}, \\alpha >2 <\/span>\n\n\n\n

Nota: La variable aleatoria Pareto es el ejemplo mas destacado de variables aleatorias con colas largas y puede ser vista como una versi\u00f3n cont\u00ednua de la variable aleatoria discreta Zipf<\/em><\/p>\n\n\n\n

\n\n\n\n

Beta<\/h3>\n\n\n f_X(x) = \\begin{cases} \\frac{\\Gamma (\\alpha + \\beta)}{\\Gamma(\\alpha) \\Gamma(\\beta)} x^{\\alpha-1} (1-x)^{\\beta-1} && ,\\alpha>0 \\\\, && \\beta>0 \\\\ , && 0 < x < 1 \\\\ 0 && \\text{otro caso} \\end{cases} <\/span>\n\n\n E[X] = \\frac{\\alpha}{\\alpha + \\beta} <\/span>\n\n\n VAR[X] = \\frac{\\alpha\\beta}{(\\alpha + \\beta)^2 (\\alpha + \\beta + 1)} <\/span>\n\n\n\n

Nota: La variable aleatoria Beta es \u00fatil para modelar una variedad de formas de funciones de densidad de probabilidad en intervalos finitos <\/em><\/p>\n\n\n\n

\n\n\n\n
\n

uniforme<\/a><\/p>\n\n\n\n

exponencial<\/a><\/p>\n\n\n\n

Normal<\/a><\/p>\n\n\n\n

Gamma<\/a><\/p>\n\n\n\n

Erlang m-1<\/a><\/p>\n\n\n\n

Chi-cuadrado<\/a><\/p>\n\n\n\n

Lapacian<\/a><\/p>\n\n\n\n

Rayleigh<\/a><\/p>\n\n\n\n

Cauchy<\/a><\/p>\n\n\n\n

Pareto<\/a><\/p>\n\n\n\n

Beta<\/a><\/p>\n<\/div>\n\n\n\n

\n","protected":false},"excerpt":{"rendered":"

Referencia: Le\u00f3n-Garc\u00eda 445 Important Continuous Random Variables p164 uniforme exponencial Normal Gamma Erlang m-1 Chi-cuadrado Lapacian Rayleigh Cauchy Pareto Beta Uniforme Exponencial Nota: La variable aleatoria exponencial es la \u00fanica variable aleatoria cont\u00ednua con propiedad \"sin memoria\" Normal o Gausiana - \u221e < x < + \u221e , \u03c3 >0 Nota: En un amplio rango […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-stp-unidades","format":"standard","meta":{"footnotes":""},"categories":[205],"tags":[],"class_list":["post-1068","post","type-post","status-publish","format-standard","hentry","category-stp-recursos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1068","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=1068"}],"version-history":[{"count":7,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1068\/revisions"}],"predecessor-version":[{"id":24343,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1068\/revisions\/24343"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=1068"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=1068"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=1068"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}