What does $p = P(X_1 X_3 < X_4)$ mean?
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
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.
probability
add a comment |Â
up vote
1
down vote
favorite
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.
probability
Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
â 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 $X_3 < X_4$".
â GEdgar
Dec 27 '17 at 13:41
Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
â MartÃn 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.
â Jack
Dec 27 '17 at 13:45
@Jack Yes. 11th Edition
â evergreenhomeland
Dec 27 '17 at 13:49
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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.
probability
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.
probability
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 $X_2 > X_3$.
â 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 $X_3 < X_4$".
â GEdgar
Dec 27 '17 at 13:41
Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
â MartÃn 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.
â Jack
Dec 27 '17 at 13:45
@Jack Yes. 11th Edition
â evergreenhomeland
Dec 27 '17 at 13:49
add a comment |Â
Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
â 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 $X_3 < X_4$".
â GEdgar
Dec 27 '17 at 13:41
Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
â MartÃn 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.
â Jack
Dec 27 '17 at 13:45
@Jack Yes. 11th Edition
â 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 $X_2 > X_3$.
â 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 $X_2 > X_3$.
â 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 $X_3 < X_4$".
â 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 $X_3 < X_4$".
â GEdgar
Dec 27 '17 at 13:41
Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
â MartÃn Vacas Vignolo
Dec 27 '17 at 13:42
Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
â MartÃn 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.
â 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.
â Jack
Dec 27 '17 at 13:45
@Jack Yes. 11th Edition
â evergreenhomeland
Dec 27 '17 at 13:49
@Jack Yes. 11th Edition
â evergreenhomeland
Dec 27 '17 at 13:49
add a comment |Â
2 Answers
2
active
oldest
votes
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:
Here is the official solution by Ross:
Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf
Thanks. How do you know F(X_i) is a uniform distribution function?
â evergreenhomeland
Dec 27 '17 at 16:53
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
add a comment |Â
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.
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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:
Here is the official solution by Ross:
Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf
Thanks. How do you know F(X_i) is a uniform distribution function?
â evergreenhomeland
Dec 27 '17 at 16:53
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
add a comment |Â
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:
Here is the official solution by Ross:
Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf
Thanks. How do you know F(X_i) is a uniform distribution function?
â evergreenhomeland
Dec 27 '17 at 16:53
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
add a comment |Â
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:
Here is the official solution by Ross:
Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf
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:
Here is the official solution by Ross:
Thanks to http://math2.uncc.edu/~imsonin/Ross_Probability10ed_Student_Solutions2010.pdf
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?
â evergreenhomeland
Dec 27 '17 at 16:53
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
add a comment |Â
Thanks. How do you know F(X_i) is a uniform distribution function?
â evergreenhomeland
Dec 27 '17 at 16:53
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
Thanks. How do you know F(X_i) is a uniform distribution function?
â evergreenhomeland
Dec 27 '17 at 16:53
Thanks. How do you know F(X_i) is a uniform distribution function?
â evergreenhomeland
Dec 27 '17 at 16:53
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
@evergreenhomeland: this is a very instructive exercise. Try it!
â Jack
Dec 27 '17 at 17:09
add a comment |Â
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.
add a comment |Â
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.
add a comment |Â
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.
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.
answered Dec 27 '17 at 13:42
stuart stevenson
3971213
3971213
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2581816%2fwhat-does-p-px-1-x-2-x-3-x-4-mean%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Usually, a "multiple" inequality is a conjucntion: $X_1 < X_2 > X_3$ is $X_1 < X_2$ and $X_2 > X_3$.
â 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 $X_3 < X_4$".
â GEdgar
Dec 27 '17 at 13:41
Or maybe a typo and is $mathbbP(X_1<X_2<X_3<X_4)$
â MartÃn 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.
â Jack
Dec 27 '17 at 13:45
@Jack Yes. 11th Edition
â evergreenhomeland
Dec 27 '17 at 13:49