Probability of an equation being less than/greater than a certain value
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I'm studying probability and the form of a problem that we are working on is:
$x^2 + y^2 < n/k$, the question being what is the probability that the equation is less than (or greater than) the $n/k$, given intervals for $x$ and $y$ such as
$0 < x <1$, $-1 < y < 1$ and $n$ and $k$ are just two random values.
$n$ and $y$ are just chosen values. Distribution is not specified
A concrete example:
What is the probability that $x^2 + (1-y)^2 < 1/9$
I had started using joint density functions and integrating with respect to $x$ and $y$ using the intervals as the support for the integration limits. Doing that gets me $8/9$ but I'm not certain if that is indicative of $8/9$ < $1/9$, so the probability is just $0$ or what??? :) Any help is greatly appreciated.
Thanks!
probability
 |Â
show 2 more comments
up vote
1
down vote
favorite
I'm studying probability and the form of a problem that we are working on is:
$x^2 + y^2 < n/k$, the question being what is the probability that the equation is less than (or greater than) the $n/k$, given intervals for $x$ and $y$ such as
$0 < x <1$, $-1 < y < 1$ and $n$ and $k$ are just two random values.
$n$ and $y$ are just chosen values. Distribution is not specified
A concrete example:
What is the probability that $x^2 + (1-y)^2 < 1/9$
I had started using joint density functions and integrating with respect to $x$ and $y$ using the intervals as the support for the integration limits. Doing that gets me $8/9$ but I'm not certain if that is indicative of $8/9$ < $1/9$, so the probability is just $0$ or what??? :) Any help is greatly appreciated.
Thanks!
probability
1
What do you mean by random values? What is the distribution of $x, y, n,k$, and are they chosen independently? Also, in the example, you specified an $n$ and a $k$. So are they fixed? Because then they are not random. Be more specific.
– A. Pongrácz
Sep 7 at 5:51
You need to calculate the circles with radius less than $frac13$ centered at $(0,1)$ in the given interval.
– prog_SAHIL
Sep 7 at 6:07
"Distribution is not specified" Which is BAD but probably means one is assuming uniform distributions. (Odd to see an answer avoiding this crucial point, that you are explicitely asking about, being accepted after 5 minutes.)
– Did
Sep 7 at 6:40
@Did, How is one going to solve this without specified distribution. Uniform distribution is the way to go in such cases.
– prog_SAHIL
Sep 7 at 6:44
@prog_SAHIL Indeed one needs a distribution to be specified, and this is not the case here, which makes this exercise a badly composed and misleading one, most probably by sheer laziness of the person who wrote it. "Uniform distribution is the way to go in such cases." Certainly not by hiding the problem under the rug, as your answer does.
– Did
Sep 7 at 6:48
 |Â
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm studying probability and the form of a problem that we are working on is:
$x^2 + y^2 < n/k$, the question being what is the probability that the equation is less than (or greater than) the $n/k$, given intervals for $x$ and $y$ such as
$0 < x <1$, $-1 < y < 1$ and $n$ and $k$ are just two random values.
$n$ and $y$ are just chosen values. Distribution is not specified
A concrete example:
What is the probability that $x^2 + (1-y)^2 < 1/9$
I had started using joint density functions and integrating with respect to $x$ and $y$ using the intervals as the support for the integration limits. Doing that gets me $8/9$ but I'm not certain if that is indicative of $8/9$ < $1/9$, so the probability is just $0$ or what??? :) Any help is greatly appreciated.
Thanks!
probability
I'm studying probability and the form of a problem that we are working on is:
$x^2 + y^2 < n/k$, the question being what is the probability that the equation is less than (or greater than) the $n/k$, given intervals for $x$ and $y$ such as
$0 < x <1$, $-1 < y < 1$ and $n$ and $k$ are just two random values.
$n$ and $y$ are just chosen values. Distribution is not specified
A concrete example:
What is the probability that $x^2 + (1-y)^2 < 1/9$
I had started using joint density functions and integrating with respect to $x$ and $y$ using the intervals as the support for the integration limits. Doing that gets me $8/9$ but I'm not certain if that is indicative of $8/9$ < $1/9$, so the probability is just $0$ or what??? :) Any help is greatly appreciated.
Thanks!
probability
probability
edited Sep 7 at 6:01
asked Sep 7 at 5:43
user590816
1
What do you mean by random values? What is the distribution of $x, y, n,k$, and are they chosen independently? Also, in the example, you specified an $n$ and a $k$. So are they fixed? Because then they are not random. Be more specific.
– A. Pongrácz
Sep 7 at 5:51
You need to calculate the circles with radius less than $frac13$ centered at $(0,1)$ in the given interval.
– prog_SAHIL
Sep 7 at 6:07
"Distribution is not specified" Which is BAD but probably means one is assuming uniform distributions. (Odd to see an answer avoiding this crucial point, that you are explicitely asking about, being accepted after 5 minutes.)
– Did
Sep 7 at 6:40
@Did, How is one going to solve this without specified distribution. Uniform distribution is the way to go in such cases.
– prog_SAHIL
Sep 7 at 6:44
@prog_SAHIL Indeed one needs a distribution to be specified, and this is not the case here, which makes this exercise a badly composed and misleading one, most probably by sheer laziness of the person who wrote it. "Uniform distribution is the way to go in such cases." Certainly not by hiding the problem under the rug, as your answer does.
– Did
Sep 7 at 6:48
 |Â
show 2 more comments
1
What do you mean by random values? What is the distribution of $x, y, n,k$, and are they chosen independently? Also, in the example, you specified an $n$ and a $k$. So are they fixed? Because then they are not random. Be more specific.
– A. Pongrácz
Sep 7 at 5:51
You need to calculate the circles with radius less than $frac13$ centered at $(0,1)$ in the given interval.
– prog_SAHIL
Sep 7 at 6:07
"Distribution is not specified" Which is BAD but probably means one is assuming uniform distributions. (Odd to see an answer avoiding this crucial point, that you are explicitely asking about, being accepted after 5 minutes.)
– Did
Sep 7 at 6:40
@Did, How is one going to solve this without specified distribution. Uniform distribution is the way to go in such cases.
– prog_SAHIL
Sep 7 at 6:44
@prog_SAHIL Indeed one needs a distribution to be specified, and this is not the case here, which makes this exercise a badly composed and misleading one, most probably by sheer laziness of the person who wrote it. "Uniform distribution is the way to go in such cases." Certainly not by hiding the problem under the rug, as your answer does.
– Did
Sep 7 at 6:48
1
1
What do you mean by random values? What is the distribution of $x, y, n,k$, and are they chosen independently? Also, in the example, you specified an $n$ and a $k$. So are they fixed? Because then they are not random. Be more specific.
– A. Pongrácz
Sep 7 at 5:51
What do you mean by random values? What is the distribution of $x, y, n,k$, and are they chosen independently? Also, in the example, you specified an $n$ and a $k$. So are they fixed? Because then they are not random. Be more specific.
– A. Pongrácz
Sep 7 at 5:51
You need to calculate the circles with radius less than $frac13$ centered at $(0,1)$ in the given interval.
– prog_SAHIL
Sep 7 at 6:07
You need to calculate the circles with radius less than $frac13$ centered at $(0,1)$ in the given interval.
– prog_SAHIL
Sep 7 at 6:07
"Distribution is not specified" Which is BAD but probably means one is assuming uniform distributions. (Odd to see an answer avoiding this crucial point, that you are explicitely asking about, being accepted after 5 minutes.)
– Did
Sep 7 at 6:40
"Distribution is not specified" Which is BAD but probably means one is assuming uniform distributions. (Odd to see an answer avoiding this crucial point, that you are explicitely asking about, being accepted after 5 minutes.)
– Did
Sep 7 at 6:40
@Did, How is one going to solve this without specified distribution. Uniform distribution is the way to go in such cases.
– prog_SAHIL
Sep 7 at 6:44
@Did, How is one going to solve this without specified distribution. Uniform distribution is the way to go in such cases.
– prog_SAHIL
Sep 7 at 6:44
@prog_SAHIL Indeed one needs a distribution to be specified, and this is not the case here, which makes this exercise a badly composed and misleading one, most probably by sheer laziness of the person who wrote it. "Uniform distribution is the way to go in such cases." Certainly not by hiding the problem under the rug, as your answer does.
– Did
Sep 7 at 6:48
@prog_SAHIL Indeed one needs a distribution to be specified, and this is not the case here, which makes this exercise a badly composed and misleading one, most probably by sheer laziness of the person who wrote it. "Uniform distribution is the way to go in such cases." Certainly not by hiding the problem under the rug, as your answer does.
– Did
Sep 7 at 6:48
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
As question specifies no distribution, I am assuming uniform
distribution.As @Did pointed out this is a badly composed problem, but
since this answer solves a part of the problem. I will keep this
intact.
You are looking at equation of an circle with restricted domain, This can be easily plotted as,
The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2ltfrac19$$ is
$$frac14pifrac13^2=fracpi36$$
The area of the domain is $2cdot1=2$.
Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$fracfracpi362=fracpi72$$
edited Sep 7 at 7:42
Did
243k23209444
answered Sep 7 at 6:16
prog_SAHIL
1,517318
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
As question specifies no distribution, I am assuming uniform
distribution.As @Did pointed out this is a badly composed problem, but
since this answer solves a part of the problem. I will keep this
intact.
You are looking at equation of an circle with restricted domain, This can be easily plotted as,
The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2ltfrac19$$ is
$$frac14pifrac13^2=fracpi36$$
The area of the domain is $2cdot1=2$.
Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$fracfracpi362=fracpi72$$
edited Sep 7 at 7:42
Did
243k23209444
answered Sep 7 at 6:16
prog_SAHIL
1,517318
add a comment |Â
up vote
1
down vote
accepted
As question specifies no distribution, I am assuming uniform
distribution.As @Did pointed out this is a badly composed problem, but
since this answer solves a part of the problem. I will keep this
intact.
You are looking at equation of an circle with restricted domain, This can be easily plotted as,
The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2ltfrac19$$ is
$$frac14pifrac13^2=fracpi36$$
The area of the domain is $2cdot1=2$.
Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$fracfracpi362=fracpi72$$
edited Sep 7 at 7:42
Did
243k23209444
answered Sep 7 at 6:16
prog_SAHIL
1,517318
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
As question specifies no distribution, I am assuming uniform
distribution.As @Did pointed out this is a badly composed problem, but
since this answer solves a part of the problem. I will keep this
intact.
You are looking at equation of an circle with restricted domain, This can be easily plotted as,
The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2ltfrac19$$ is
$$frac14pifrac13^2=fracpi36$$
The area of the domain is $2cdot1=2$.
Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$fracfracpi362=fracpi72$$
edited Sep 7 at 7:42
Did
243k23209444
answered Sep 7 at 6:16
prog_SAHIL
1,517318
As question specifies no distribution, I am assuming uniform
distribution.As @Did pointed out this is a badly composed problem, but
since this answer solves a part of the problem. I will keep this
intact.
You are looking at equation of an circle with restricted domain, This can be easily plotted as,
The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2ltfrac19$$ is
$$frac14pifrac13^2=fracpi36$$
The area of the domain is $2cdot1=2$.
Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$fracfracpi362=fracpi72$$
edited Sep 7 at 7:42
Did
243k23209444
answered Sep 7 at 6:16
prog_SAHIL
1,517318
edited Sep 7 at 7:42
Did
243k23209444
edited Sep 7 at 7:42
Did
243k23209444
edited Sep 7 at 7:42
Did
243k23209444
243k23209444
answered Sep 7 at 6:16
prog_SAHIL
1,517318
answered Sep 7 at 6:16
prog_SAHIL
1,517318
answered Sep 7 at 6:16
prog_SAHIL
1,517318
1,517318
add a comment |Â
add a comment |Â
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1
What do you mean by random values? What is the distribution of $x, y, n,k$, and are they chosen independently? Also, in the example, you specified an $n$ and a $k$. So are they fixed? Because then they are not random. Be more specific.
– A. Pongrácz
Sep 7 at 5:51
You need to calculate the circles with radius less than $frac13$ centered at $(0,1)$ in the given interval.
– prog_SAHIL
Sep 7 at 6:07
"Distribution is not specified" Which is BAD but probably means one is assuming uniform distributions. (Odd to see an answer avoiding this crucial point, that you are explicitely asking about, being accepted after 5 minutes.)
– Did
Sep 7 at 6:40
@Did, How is one going to solve this without specified distribution. Uniform distribution is the way to go in such cases.
– prog_SAHIL
Sep 7 at 6:44
@prog_SAHIL Indeed one needs a distribution to be specified, and this is not the case here, which makes this exercise a badly composed and misleading one, most probably by sheer laziness of the person who wrote it. "Uniform distribution is the way to go in such cases." Certainly not by hiding the problem under the rug, as your answer does.
– Did
Sep 7 at 6:48