Probability of points on a Line
up vote
0
down vote
favorite
Three points $X_1,X_2,X_3$ are selected at random on a line of length $L$. What is the probability that $X_2$ lies between $X_1$ $and$ $X_3$?
I know that all three are equally likely to be in the middle, but I don’t know how that is so, I’m looking for a way to come to that answer.
probability probability-theory probability-distributions random-variables
asked Dec 1 at 21:25
user601297
895
add a comment |
up vote
0
down vote
favorite
Three points $X_1,X_2,X_3$ are selected at random on a line of length $L$. What is the probability that $X_2$ lies between $X_1$ $and$ $X_3$?
I know that all three are equally likely to be in the middle, but I don’t know how that is so, I’m looking for a way to come to that answer.
probability probability-theory probability-distributions random-variables
asked Dec 1 at 21:25
user601297
895
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Three points $X_1,X_2,X_3$ are selected at random on a line of length $L$. What is the probability that $X_2$ lies between $X_1$ $and$ $X_3$?
I know that all three are equally likely to be in the middle, but I don’t know how that is so, I’m looking for a way to come to that answer.
probability probability-theory probability-distributions random-variables
asked Dec 1 at 21:25
user601297
895
Three points $X_1,X_2,X_3$ are selected at random on a line of length $L$. What is the probability that $X_2$ lies between $X_1$ $and$ $X_3$?
I know that all three are equally likely to be in the middle, but I don’t know how that is so, I’m looking for a way to come to that answer.
probability probability-theory probability-distributions random-variables
probability probability-theory probability-distributions random-variables
asked Dec 1 at 21:25
user601297
895
asked Dec 1 at 21:25
user601297
895
asked Dec 1 at 21:25
user601297
895
asked Dec 1 at 21:25
user601297
895
asked Dec 1 at 21:25
user601297
895
895
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
Clearly $1/3$ due to permutation symmetry.
answered Dec 1 at 21:44
David G. Stork
9,31921232
add a comment |
up vote
0
down vote
Assuming that $X_{i}$ is coming from a continuous uniform distribution on the interval $(0,L)$ then $P(X_{i}=a)=0$, for any $a$ on the real line. Note that there are many ways to construct an interval of length $L$, such as $(-frac{L}{2},frac{L}{2}$), but I will assume in this case, the interval is $(0,L)$. Then your question is asking to find:
$P(X_{3}<X_{2}<X_{1})$
=$int_{x_{3}=0}^{L} int_{x_{3}}^{L}int_{x_{2}}^{L} frac{1}{L^{3}}dx_{1}dx_{2}dx_{3}$
answered Dec 1 at 22:12
Live Free or π Hard
440213
1
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Clearly $1/3$ due to permutation symmetry.
answered Dec 1 at 21:44
David G. Stork
9,31921232
add a comment |
up vote
1
down vote
Clearly $1/3$ due to permutation symmetry.
answered Dec 1 at 21:44
David G. Stork
9,31921232
add a comment |
up vote
1
down vote
up vote
1
down vote
Clearly $1/3$ due to permutation symmetry.
answered Dec 1 at 21:44
David G. Stork
9,31921232
Clearly $1/3$ due to permutation symmetry.
answered Dec 1 at 21:44
David G. Stork
9,31921232
answered Dec 1 at 21:44
David G. Stork
9,31921232
answered Dec 1 at 21:44
David G. Stork
9,31921232
answered Dec 1 at 21:44
David G. Stork
9,31921232
9,31921232
add a comment |
add a comment |
up vote
0
down vote
Assuming that $X_{i}$ is coming from a continuous uniform distribution on the interval $(0,L)$ then $P(X_{i}=a)=0$, for any $a$ on the real line. Note that there are many ways to construct an interval of length $L$, such as $(-frac{L}{2},frac{L}{2}$), but I will assume in this case, the interval is $(0,L)$. Then your question is asking to find:
$P(X_{3}<X_{2}<X_{1})$
=$int_{x_{3}=0}^{L} int_{x_{3}}^{L}int_{x_{2}}^{L} frac{1}{L^{3}}dx_{1}dx_{2}dx_{3}$
answered Dec 1 at 22:12
Live Free or π Hard
440213
1
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
add a comment |
up vote
0
down vote
Assuming that $X_{i}$ is coming from a continuous uniform distribution on the interval $(0,L)$ then $P(X_{i}=a)=0$, for any $a$ on the real line. Note that there are many ways to construct an interval of length $L$, such as $(-frac{L}{2},frac{L}{2}$), but I will assume in this case, the interval is $(0,L)$. Then your question is asking to find:
$P(X_{3}<X_{2}<X_{1})$
=$int_{x_{3}=0}^{L} int_{x_{3}}^{L}int_{x_{2}}^{L} frac{1}{L^{3}}dx_{1}dx_{2}dx_{3}$
answered Dec 1 at 22:12
Live Free or π Hard
440213
1
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
add a comment |
up vote
0
down vote
up vote
0
down vote
Assuming that $X_{i}$ is coming from a continuous uniform distribution on the interval $(0,L)$ then $P(X_{i}=a)=0$, for any $a$ on the real line. Note that there are many ways to construct an interval of length $L$, such as $(-frac{L}{2},frac{L}{2}$), but I will assume in this case, the interval is $(0,L)$. Then your question is asking to find:
$P(X_{3}<X_{2}<X_{1})$
=$int_{x_{3}=0}^{L} int_{x_{3}}^{L}int_{x_{2}}^{L} frac{1}{L^{3}}dx_{1}dx_{2}dx_{3}$
answered Dec 1 at 22:12
Live Free or π Hard
440213
Assuming that $X_{i}$ is coming from a continuous uniform distribution on the interval $(0,L)$ then $P(X_{i}=a)=0$, for any $a$ on the real line. Note that there are many ways to construct an interval of length $L$, such as $(-frac{L}{2},frac{L}{2}$), but I will assume in this case, the interval is $(0,L)$. Then your question is asking to find:
$P(X_{3}<X_{2}<X_{1})$
=$int_{x_{3}=0}^{L} int_{x_{3}}^{L}int_{x_{2}}^{L} frac{1}{L^{3}}dx_{1}dx_{2}dx_{3}$
answered Dec 1 at 22:12
Live Free or π Hard
440213
edited Dec 1 at 22:19
answered Dec 1 at 22:12
Live Free or π Hard
440213
answered Dec 1 at 22:12
Live Free or π Hard
440213
answered Dec 1 at 22:12
Live Free or π Hard
440213
440213
1
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
add a comment |
1
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
1
1
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
in the integral, it should be $frac{1}{L^3}$ instead of $frac{1}{L}$.
– Daniel
Dec 1 at 22:15
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
@ Daniel...thank you. Yes, it’s the joint density function, again assuming that all RVs are independent.
– Live Free or π Hard
Dec 1 at 22:18
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3021850%2fprobability-of-points-on-a-line%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
