What does $p = P(X_1 X_3 < X_4)$ mean?

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It is a question 59 on page 87 from Ross's book (Introduction to Probability Models)



Let $X_1,X_2,X_3,X_4$ are independent continuous random variables with a common distribution function F and let



$p = P(X_1 < X_2 > X_3 < X_4)$



Just as the Title, what does it mean?
Or similar questions with such an inequity?



Thanks



Update 1:



The solution says:
Use the fact that F(Xi) is a uniform (0,1) random variable to obtain.
But where is this fact?



Update: A similar question



How can I compute an expression for $P(X_1>X_2>X_3>X_4)$ if $X_1,X_2,X_3,X_4$ are normal and mututally independent?



BTW



I am not a native-English speaker, and I am learning it by myself.







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  • Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
    – Mauro ALLEGRANZA
    Dec 27 '17 at 13:41











  • Of course "$X_1 < X_2 > X_3 < X_4$" means "$X_1 < X_2$ and $X_2 > X_3$ and $X_3 < X_4$".
    – GEdgar
    Dec 27 '17 at 13:41










  • Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
    – Martín Vacas Vignolo
    Dec 27 '17 at 13:42











  • Are you sure that is the correct name of the book? Ross has a book called "Introduction to Probability Models" and there are many different editions.
    – Jack
    Dec 27 '17 at 13:45











  • @Jack Yes. 11th Edition
    – evergreenhomeland
    Dec 27 '17 at 13:49














up vote
1
down vote

favorite
1












It is a question 59 on page 87 from Ross's book (Introduction to Probability Models)



Let $X_1,X_2,X_3,X_4$ are independent continuous random variables with a common distribution function F and let



$p = P(X_1 < X_2 > X_3 < X_4)$



Just as the Title, what does it mean?
Or similar questions with such an inequity?



Thanks



Update 1:



The solution says:
Use the fact that F(Xi) is a uniform (0,1) random variable to obtain.
But where is this fact?



Update: A similar question



How can I compute an expression for $P(X_1>X_2>X_3>X_4)$ if $X_1,X_2,X_3,X_4$ are normal and mututally independent?



BTW



I am not a native-English speaker, and I am learning it by myself.







share|cite|improve this question






















  • Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
    – Mauro ALLEGRANZA
    Dec 27 '17 at 13:41











  • Of course "$X_1 < X_2 > X_3 < X_4$" means "$X_1 < X_2$ and $X_2 > X_3$ and $X_3 < X_4$".
    – GEdgar
    Dec 27 '17 at 13:41










  • Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
    – Martín Vacas Vignolo
    Dec 27 '17 at 13:42











  • Are you sure that is the correct name of the book? Ross has a book called "Introduction to Probability Models" and there are many different editions.
    – Jack
    Dec 27 '17 at 13:45











  • @Jack Yes. 11th Edition
    – evergreenhomeland
    Dec 27 '17 at 13:49












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





It is a question 59 on page 87 from Ross's book (Introduction to Probability Models)



Let $X_1,X_2,X_3,X_4$ are independent continuous random variables with a common distribution function F and let



$p = P(X_1 < X_2 > X_3 < X_4)$



Just as the Title, what does it mean?
Or similar questions with such an inequity?



Thanks



Update 1:



The solution says:
Use the fact that F(Xi) is a uniform (0,1) random variable to obtain.
But where is this fact?



Update: A similar question



How can I compute an expression for $P(X_1>X_2>X_3>X_4)$ if $X_1,X_2,X_3,X_4$ are normal and mututally independent?



BTW



I am not a native-English speaker, and I am learning it by myself.







share|cite|improve this question














It is a question 59 on page 87 from Ross's book (Introduction to Probability Models)



Let $X_1,X_2,X_3,X_4$ are independent continuous random variables with a common distribution function F and let



$p = P(X_1 < X_2 > X_3 < X_4)$



Just as the Title, what does it mean?
Or similar questions with such an inequity?



Thanks



Update 1:



The solution says:
Use the fact that F(Xi) is a uniform (0,1) random variable to obtain.
But where is this fact?



Update: A similar question



How can I compute an expression for $P(X_1>X_2>X_3>X_4)$ if $X_1,X_2,X_3,X_4$ are normal and mututally independent?



BTW



I am not a native-English speaker, and I am learning it by myself.









share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 24 at 9:01

























asked Dec 27 '17 at 13:37









evergreenhomeland

828




828











  • Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
    – Mauro ALLEGRANZA
    Dec 27 '17 at 13:41











  • Of course "$X_1 < X_2 > X_3 < X_4$" means "$X_1 < X_2$ and $X_2 > X_3$ and $X_3 < X_4$".
    – GEdgar
    Dec 27 '17 at 13:41










  • Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
    – Martín Vacas Vignolo
    Dec 27 '17 at 13:42











  • Are you sure that is the correct name of the book? Ross has a book called "Introduction to Probability Models" and there are many different editions.
    – Jack
    Dec 27 '17 at 13:45











  • @Jack Yes. 11th Edition
    – evergreenhomeland
    Dec 27 '17 at 13:49
















  • Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
    – Mauro ALLEGRANZA
    Dec 27 '17 at 13:41











  • Of course "$X_1 < X_2 > X_3 < X_4$" means "$X_1 < X_2$ and $X_2 > X_3$ and $X_3 < X_4$".
    – GEdgar
    Dec 27 '17 at 13:41










  • Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
    – Martín Vacas Vignolo
    Dec 27 '17 at 13:42











  • Are you sure that is the correct name of the book? Ross has a book called "Introduction to Probability Models" and there are many different editions.
    – Jack
    Dec 27 '17 at 13:45











  • @Jack Yes. 11th Edition
    – evergreenhomeland
    Dec 27 '17 at 13:49















Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
– Mauro ALLEGRANZA
Dec 27 '17 at 13:41





Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
– Mauro ALLEGRANZA
Dec 27 '17 at 13:41













Of course "$X_1 < X_2 > X_3 < X_4$" means "$X_1 < X_2$ and $X_2 > X_3$ and $X_3 < X_4$".
– GEdgar
Dec 27 '17 at 13:41




Of course "$X_1 < X_2 > X_3 < X_4$" means "$X_1 < X_2$ and $X_2 > X_3$ and $X_3 < X_4$".
– GEdgar
Dec 27 '17 at 13:41












Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
– Martín Vacas Vignolo
Dec 27 '17 at 13:42





Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
– Martín Vacas Vignolo
Dec 27 '17 at 13:42













Are you sure that is the correct name of the book? Ross has a book called "Introduction to Probability Models" and there are many different editions.
– Jack
Dec 27 '17 at 13:45





Are you sure that is the correct name of the book? Ross has a book called "Introduction to Probability Models" and there are many different editions.
– Jack
Dec 27 '17 at 13:45













@Jack Yes. 11th Edition
– evergreenhomeland
Dec 27 '17 at 13:49




@Jack Yes. 11th Edition
– evergreenhomeland
Dec 27 '17 at 13:49










2 Answers
2






active

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up vote
0
down vote



accepted










This seems to be a rather unconventional use of inequalities as GEdgar points out in his comment and there is no typo. The "chain" of inequalities $X_1<X_2>X_3<X_4$ means all of the following hold:
$$
X_1<X_2, X_2>X_3, X_3<X_4.
$$



Here is the original problem:




enter image description here




Here is the official solution by Ross:




enter image description here




Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf






share|cite|improve this answer






















  • Thanks. How do you know F(X_i) is a uniform distribution function?
    – evergreenhomeland
    Dec 27 '17 at 16:53










  • @evergreenhomeland: this is a very instructive exercise. Try it!
    – Jack
    Dec 27 '17 at 17:09

















up vote
1
down vote













It's the probability that $X_1$ is less than $X_2$ AND that $X_2$ is greater than $X_3$ AND that $X_3$ is less than $X_4$. That's all.






share|cite|improve this answer




















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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

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    active

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    up vote
    0
    down vote



    accepted










    This seems to be a rather unconventional use of inequalities as GEdgar points out in his comment and there is no typo. The "chain" of inequalities $X_1<X_2>X_3<X_4$ means all of the following hold:
    $$
    X_1<X_2, X_2>X_3, X_3<X_4.
    $$



    Here is the original problem:




    enter image description here




    Here is the official solution by Ross:




    enter image description here




    Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf






    share|cite|improve this answer






















    • Thanks. How do you know F(X_i) is a uniform distribution function?
      – evergreenhomeland
      Dec 27 '17 at 16:53










    • @evergreenhomeland: this is a very instructive exercise. Try it!
      – Jack
      Dec 27 '17 at 17:09














    up vote
    0
    down vote



    accepted










    This seems to be a rather unconventional use of inequalities as GEdgar points out in his comment and there is no typo. The "chain" of inequalities $X_1<X_2>X_3<X_4$ means all of the following hold:
    $$
    X_1<X_2, X_2>X_3, X_3<X_4.
    $$



    Here is the original problem:




    enter image description here




    Here is the official solution by Ross:




    enter image description here




    Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf






    share|cite|improve this answer






















    • Thanks. How do you know F(X_i) is a uniform distribution function?
      – evergreenhomeland
      Dec 27 '17 at 16:53










    • @evergreenhomeland: this is a very instructive exercise. Try it!
      – Jack
      Dec 27 '17 at 17:09












    up vote
    0
    down vote



    accepted







    up vote
    0
    down vote



    accepted






    This seems to be a rather unconventional use of inequalities as GEdgar points out in his comment and there is no typo. The "chain" of inequalities $X_1<X_2>X_3<X_4$ means all of the following hold:
    $$
    X_1<X_2, X_2>X_3, X_3<X_4.
    $$



    Here is the original problem:




    enter image description here




    Here is the official solution by Ross:




    enter image description here




    Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf






    share|cite|improve this answer














    This seems to be a rather unconventional use of inequalities as GEdgar points out in his comment and there is no typo. The "chain" of inequalities $X_1<X_2>X_3<X_4$ means all of the following hold:
    $$
    X_1<X_2, X_2>X_3, X_3<X_4.
    $$



    Here is the original problem:




    enter image description here




    Here is the official solution by Ross:




    enter image description here




    Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 27 '17 at 14:39

























    answered Dec 27 '17 at 14:25









    Jack

    26.6k1678192




    26.6k1678192











    • Thanks. How do you know F(X_i) is a uniform distribution function?
      – evergreenhomeland
      Dec 27 '17 at 16:53










    • @evergreenhomeland: this is a very instructive exercise. Try it!
      – Jack
      Dec 27 '17 at 17:09
















    • Thanks. How do you know F(X_i) is a uniform distribution function?
      – evergreenhomeland
      Dec 27 '17 at 16:53










    • @evergreenhomeland: this is a very instructive exercise. Try it!
      – Jack
      Dec 27 '17 at 17:09















    Thanks. How do you know F(X_i) is a uniform distribution function?
    – evergreenhomeland
    Dec 27 '17 at 16:53




    Thanks. How do you know F(X_i) is a uniform distribution function?
    – evergreenhomeland
    Dec 27 '17 at 16:53












    @evergreenhomeland: this is a very instructive exercise. Try it!
    – Jack
    Dec 27 '17 at 17:09




    @evergreenhomeland: this is a very instructive exercise. Try it!
    – Jack
    Dec 27 '17 at 17:09










    up vote
    1
    down vote













    It's the probability that $X_1$ is less than $X_2$ AND that $X_2$ is greater than $X_3$ AND that $X_3$ is less than $X_4$. That's all.






    share|cite|improve this answer
























      up vote
      1
      down vote













      It's the probability that $X_1$ is less than $X_2$ AND that $X_2$ is greater than $X_3$ AND that $X_3$ is less than $X_4$. That's all.






      share|cite|improve this answer






















        up vote
        1
        down vote










        up vote
        1
        down vote









        It's the probability that $X_1$ is less than $X_2$ AND that $X_2$ is greater than $X_3$ AND that $X_3$ is less than $X_4$. That's all.






        share|cite|improve this answer












        It's the probability that $X_1$ is less than $X_2$ AND that $X_2$ is greater than $X_3$ AND that $X_3$ is less than $X_4$. That's all.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 27 '17 at 13:42









        stuart stevenson

        3971213




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