Independence between random vector and event











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Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables.



Let $X_1,...,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables.



Consider that $X_i$ and $U_j$ are independents, for all $i,j$.



Show that the event ${U_1 > U_2 > U_3}$ is independent of the $(U_{(3)},X_{(1)})$, where $U_{(3)} = max{U_1,U_2,U_3}$ and $X_{(1)} = min{X_1,...,X_n}$.




I have no idea to start. How'd be the definition of independence between events and random variables?






probability probability-theory probability-distributions independence uniform-distribution







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    As for your last question: the rv. $X$ is independent of an event $A$ if all events $B$ defined in terms of $X$ are independent of $A$, such as $B=[Xlt t]$, or $B=[Xin S]$, or most generally, for all $B$ in the sigma field $mathcal{F}(X)$ generated by $X$
    – kimchi lover
    Nov 28 at 22:40








  • 1




    You also need independence between $U_k$ and$X_j$ for all indices.
    – herb steinberg
    Nov 28 at 22:46















up vote
1
down vote

favorite
2













Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables.



Let $X_1,...,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables.



Consider that $X_i$ and $U_j$ are independents, for all $i,j$.



Show that the event ${U_1 > U_2 > U_3}$ is independent of the $(U_{(3)},X_{(1)})$, where $U_{(3)} = max{U_1,U_2,U_3}$ and $X_{(1)} = min{X_1,...,X_n}$.




I have no idea to start. How'd be the definition of independence between events and random variables?






probability probability-theory probability-distributions independence uniform-distribution







share|cite|improve this question




















  • 1




    As for your last question: the rv. $X$ is independent of an event $A$ if all events $B$ defined in terms of $X$ are independent of $A$, such as $B=[Xlt t]$, or $B=[Xin S]$, or most generally, for all $B$ in the sigma field $mathcal{F}(X)$ generated by $X$
    – kimchi lover
    Nov 28 at 22:40








  • 1




    You also need independence between $U_k$ and$X_j$ for all indices.
    – herb steinberg
    Nov 28 at 22:46













up vote
1
down vote

favorite
2









up vote
1
down vote

favorite
2






2






Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables.



Let $X_1,...,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables.



Consider that $X_i$ and $U_j$ are independents, for all $i,j$.



Show that the event ${U_1 > U_2 > U_3}$ is independent of the $(U_{(3)},X_{(1)})$, where $U_{(3)} = max{U_1,U_2,U_3}$ and $X_{(1)} = min{X_1,...,X_n}$.




I have no idea to start. How'd be the definition of independence between events and random variables?






probability probability-theory probability-distributions independence uniform-distribution







share|cite|improve this question
















Let $U_1, U_2$ and $U_3$ be three independent uniformly $(0, 1)$ random variables.



Let $X_1,...,X_n$ be a sequence of independent uniformly $(0, 1)$ random variables.



Consider that $X_i$ and $U_j$ are independents, for all $i,j$.



Show that the event ${U_1 > U_2 > U_3}$ is independent of the $(U_{(3)},X_{(1)})$, where $U_{(3)} = max{U_1,U_2,U_3}$ and $X_{(1)} = min{X_1,...,X_n}$.




I have no idea to start. How'd be the definition of independence between events and random variables?






probability probability-theory probability-distributions independence uniform-distribution




probability probability-theory probability-distributions independence uniform-distribution






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share|cite|improve this question













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edited Nov 29 at 10:00









Davide Giraudo

124k16150259




124k16150259










asked Nov 28 at 21:18









Pedro Salgado

675




675








  • 1




    As for your last question: the rv. $X$ is independent of an event $A$ if all events $B$ defined in terms of $X$ are independent of $A$, such as $B=[Xlt t]$, or $B=[Xin S]$, or most generally, for all $B$ in the sigma field $mathcal{F}(X)$ generated by $X$
    – kimchi lover
    Nov 28 at 22:40








  • 1




    You also need independence between $U_k$ and$X_j$ for all indices.
    – herb steinberg
    Nov 28 at 22:46














  • 1




    As for your last question: the rv. $X$ is independent of an event $A$ if all events $B$ defined in terms of $X$ are independent of $A$, such as $B=[Xlt t]$, or $B=[Xin S]$, or most generally, for all $B$ in the sigma field $mathcal{F}(X)$ generated by $X$
    – kimchi lover
    Nov 28 at 22:40








  • 1




    You also need independence between $U_k$ and$X_j$ for all indices.
    – herb steinberg
    Nov 28 at 22:46








1




1




As for your last question: the rv. $X$ is independent of an event $A$ if all events $B$ defined in terms of $X$ are independent of $A$, such as $B=[Xlt t]$, or $B=[Xin S]$, or most generally, for all $B$ in the sigma field $mathcal{F}(X)$ generated by $X$
– kimchi lover
Nov 28 at 22:40






As for your last question: the rv. $X$ is independent of an event $A$ if all events $B$ defined in terms of $X$ are independent of $A$, such as $B=[Xlt t]$, or $B=[Xin S]$, or most generally, for all $B$ in the sigma field $mathcal{F}(X)$ generated by $X$
– kimchi lover
Nov 28 at 22:40






1




1




You also need independence between $U_k$ and$X_j$ for all indices.
– herb steinberg
Nov 28 at 22:46




You also need independence between $U_k$ and$X_j$ for all indices.
– herb steinberg
Nov 28 at 22:46










1 Answer
1






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up vote
2
down vote



accepted










An event and a random variable are independent if for all supported values of the random variable, the conditional expectation of the event given the variable equals the margial probability of the event.



In short you need to establish whether: $${forall uin(0;1)~forall xin(0;1):\quadmathsf P({U_1{>}U_2{>}U_3})=mathsf P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3,x{=}min{X_j}_{j=1}^n)}$$






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  • $min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
    – Pedro Salgado
    Nov 29 at 12:22








  • 1




    Use symmetry. @PedroSalgato
    – Graham Kemp
    Nov 29 at 22:37










  • how can I use? @graham-kemp
    – Pedro Salgado
    Nov 30 at 0:05






  • 1




    Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
    – Graham Kemp
    Nov 30 at 0:23











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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes








up vote
2
down vote



accepted










An event and a random variable are independent if for all supported values of the random variable, the conditional expectation of the event given the variable equals the margial probability of the event.



In short you need to establish whether: $${forall uin(0;1)~forall xin(0;1):\quadmathsf P({U_1{>}U_2{>}U_3})=mathsf P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3,x{=}min{X_j}_{j=1}^n)}$$






share|cite|improve this answer























  • $min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
    – Pedro Salgado
    Nov 29 at 12:22








  • 1




    Use symmetry. @PedroSalgato
    – Graham Kemp
    Nov 29 at 22:37










  • how can I use? @graham-kemp
    – Pedro Salgado
    Nov 30 at 0:05






  • 1




    Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
    – Graham Kemp
    Nov 30 at 0:23















up vote
2
down vote



accepted










An event and a random variable are independent if for all supported values of the random variable, the conditional expectation of the event given the variable equals the margial probability of the event.



In short you need to establish whether: $${forall uin(0;1)~forall xin(0;1):\quadmathsf P({U_1{>}U_2{>}U_3})=mathsf P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3,x{=}min{X_j}_{j=1}^n)}$$






share|cite|improve this answer























  • $min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
    – Pedro Salgado
    Nov 29 at 12:22








  • 1




    Use symmetry. @PedroSalgato
    – Graham Kemp
    Nov 29 at 22:37










  • how can I use? @graham-kemp
    – Pedro Salgado
    Nov 30 at 0:05






  • 1




    Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
    – Graham Kemp
    Nov 30 at 0:23













up vote
2
down vote



accepted







up vote
2
down vote



accepted






An event and a random variable are independent if for all supported values of the random variable, the conditional expectation of the event given the variable equals the margial probability of the event.



In short you need to establish whether: $${forall uin(0;1)~forall xin(0;1):\quadmathsf P({U_1{>}U_2{>}U_3})=mathsf P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3,x{=}min{X_j}_{j=1}^n)}$$






share|cite|improve this answer














An event and a random variable are independent if for all supported values of the random variable, the conditional expectation of the event given the variable equals the margial probability of the event.



In short you need to establish whether: $${forall uin(0;1)~forall xin(0;1):\quadmathsf P({U_1{>}U_2{>}U_3})=mathsf P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3,x{=}min{X_j}_{j=1}^n)}$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 29 at 22:35

























answered Nov 29 at 0:53









Graham Kemp

84.7k43378




84.7k43378












  • $min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
    – Pedro Salgado
    Nov 29 at 12:22








  • 1




    Use symmetry. @PedroSalgato
    – Graham Kemp
    Nov 29 at 22:37










  • how can I use? @graham-kemp
    – Pedro Salgado
    Nov 30 at 0:05






  • 1




    Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
    – Graham Kemp
    Nov 30 at 0:23


















  • $min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
    – Pedro Salgado
    Nov 29 at 12:22








  • 1




    Use symmetry. @PedroSalgato
    – Graham Kemp
    Nov 29 at 22:37










  • how can I use? @graham-kemp
    – Pedro Salgado
    Nov 30 at 0:05






  • 1




    Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
    – Graham Kemp
    Nov 30 at 0:23
















$min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
– Pedro Salgado
Nov 29 at 12:22






$min{X_j)_{j=1}^n}$ is independent of ${U_1{>}U_2{>}U_3}$. How can I prove that ${U_1{>}U_2{>}U_3}$ and $max{U_i}_{i=1}^3$ are independent, or $P({U_1{>}U_2{>}U_3}mid u{=}max{U_i}_{i=1}^3) = P({U_1{>}U_2{>}U_3})$ ?
– Pedro Salgado
Nov 29 at 12:22






1




1




Use symmetry. @PedroSalgato
– Graham Kemp
Nov 29 at 22:37




Use symmetry. @PedroSalgato
– Graham Kemp
Nov 29 at 22:37












how can I use? @graham-kemp
– Pedro Salgado
Nov 30 at 0:05




how can I use? @graham-kemp
– Pedro Salgado
Nov 30 at 0:05




1




1




Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
– Graham Kemp
Nov 30 at 0:23




Argue that that $mathsf P({U_1>U_2>U_3}mid u{=}max{U_i}_{i=1}^3)=mathsf P({U_3>U_1>U_2}mid u{=}max{U_i}_{i=1}^3)$ and so forth, by symmetry. @Pedro-Salgado
– Graham Kemp
Nov 30 at 0:23


















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