Stuck at defining the density function for a random variable
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So I have an assignment regarding probability and random variables which states the following:
The random variable Y is defined in the following statement:
- On a random way we select a point from a circle with a radius of $6$.
- The random variable $X$ is defined as the distance between the selected point and the center of the circle and $Y = 3 - X$.
- Using the following statements define the function of distribution and the density of the distribution for the random variable $Y$.
I assume since we pick the point at random and each point has a fair chance of getting picked the Y has a uniform distribution with the constant c being equaled to 1/36*PI.
However I have no idea how to implement the random variable X or the statement Y=3-X in the equation.
Can anyone help me?
probability random-variables
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
$endgroup$
add a comment |
$begingroup$
So I have an assignment regarding probability and random variables which states the following:
The random variable Y is defined in the following statement:
- On a random way we select a point from a circle with a radius of $6$.
- The random variable $X$ is defined as the distance between the selected point and the center of the circle and $Y = 3 - X$.
- Using the following statements define the function of distribution and the density of the distribution for the random variable $Y$.
I assume since we pick the point at random and each point has a fair chance of getting picked the Y has a uniform distribution with the constant c being equaled to 1/36*PI.
However I have no idea how to implement the random variable X or the statement Y=3-X in the equation.
Can anyone help me?
probability random-variables
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
$endgroup$
4
$begingroup$
Tell me if I'm being dense, but is there anything random ($X$) about the distance of a randomly selected point on the circle from its center? Or are we, in fact, talking about picking a random point from the disc?
$endgroup$
– Matthias
Dec 16 '18 at 1:39
add a comment |
$begingroup$
So I have an assignment regarding probability and random variables which states the following:
The random variable Y is defined in the following statement:
- On a random way we select a point from a circle with a radius of $6$.
- The random variable $X$ is defined as the distance between the selected point and the center of the circle and $Y = 3 - X$.
- Using the following statements define the function of distribution and the density of the distribution for the random variable $Y$.
I assume since we pick the point at random and each point has a fair chance of getting picked the Y has a uniform distribution with the constant c being equaled to 1/36*PI.
However I have no idea how to implement the random variable X or the statement Y=3-X in the equation.
Can anyone help me?
probability random-variables
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
$endgroup$
So I have an assignment regarding probability and random variables which states the following:
The random variable Y is defined in the following statement:
- On a random way we select a point from a circle with a radius of $6$.
- The random variable $X$ is defined as the distance between the selected point and the center of the circle and $Y = 3 - X$.
- Using the following statements define the function of distribution and the density of the distribution for the random variable $Y$.
I assume since we pick the point at random and each point has a fair chance of getting picked the Y has a uniform distribution with the constant c being equaled to 1/36*PI.
However I have no idea how to implement the random variable X or the statement Y=3-X in the equation.
Can anyone help me?
probability random-variables
probability random-variables
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
edited Dec 16 '18 at 1:31
Felix Marin
68.4k7109144
68.4k7109144
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
asked Dec 15 '18 at 22:10
David DanielsDavid Daniels
132
132
4
$begingroup$
Tell me if I'm being dense, but is there anything random ($X$) about the distance of a randomly selected point on the circle from its center? Or are we, in fact, talking about picking a random point from the disc?
$endgroup$
– Matthias
Dec 16 '18 at 1:39
add a comment |
4
$begingroup$
Tell me if I'm being dense, but is there anything random ($X$) about the distance of a randomly selected point on the circle from its center? Or are we, in fact, talking about picking a random point from the disc?
$endgroup$
– Matthias
Dec 16 '18 at 1:39
4
4
$begingroup$
Tell me if I'm being dense, but is there anything random ($X$) about the distance of a randomly selected point on the circle from its center? Or are we, in fact, talking about picking a random point from the disc?
$endgroup$
– Matthias
Dec 16 '18 at 1:39
$begingroup$
Tell me if I'm being dense, but is there anything random ($X$) about the distance of a randomly selected point on the circle from its center? Or are we, in fact, talking about picking a random point from the disc?
$endgroup$
– Matthias
Dec 16 '18 at 1:39
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It is true that the uniform random distribution of points inside the circle has a factor of $dfrac{1}{36pi}.$
That is, if someone has in mind a particular subset of points within the circle and that subset of points has area $A,$ then the probability that the randomly selected point will be in that particular subset is $dfrac{A}{36pi}.$
But you have been asked for a random distribution over numbers, and points are not numbers.
What is the minimum value of $X$?
Since $X$ is the distance between the selected point and the center of the circle,
if the selected point is the center of the circle then $X=0.$
You can't have a distance less than $0,$ so that's the minimum value of $X.$
What is the maximum value of $X$? Can $X = 36pi$?
If you think it can, find a point that could be selected to make $X = 36pi.$
Can $X = 7$?
Find the set of points such that $X leq 5.$ What is the area of that set?
What is the probability that $X leq 5$?
Try this for some other numbers instead of $5.$ Do you see a pattern?
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
$endgroup$
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
add a comment |
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1 Answer
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1 Answer
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$begingroup$
It is true that the uniform random distribution of points inside the circle has a factor of $dfrac{1}{36pi}.$
That is, if someone has in mind a particular subset of points within the circle and that subset of points has area $A,$ then the probability that the randomly selected point will be in that particular subset is $dfrac{A}{36pi}.$
But you have been asked for a random distribution over numbers, and points are not numbers.
What is the minimum value of $X$?
Since $X$ is the distance between the selected point and the center of the circle,
if the selected point is the center of the circle then $X=0.$
You can't have a distance less than $0,$ so that's the minimum value of $X.$
What is the maximum value of $X$? Can $X = 36pi$?
If you think it can, find a point that could be selected to make $X = 36pi.$
Can $X = 7$?
Find the set of points such that $X leq 5.$ What is the area of that set?
What is the probability that $X leq 5$?
Try this for some other numbers instead of $5.$ Do you see a pattern?
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
$endgroup$
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
add a comment |
$begingroup$
It is true that the uniform random distribution of points inside the circle has a factor of $dfrac{1}{36pi}.$
That is, if someone has in mind a particular subset of points within the circle and that subset of points has area $A,$ then the probability that the randomly selected point will be in that particular subset is $dfrac{A}{36pi}.$
But you have been asked for a random distribution over numbers, and points are not numbers.
What is the minimum value of $X$?
Since $X$ is the distance between the selected point and the center of the circle,
if the selected point is the center of the circle then $X=0.$
You can't have a distance less than $0,$ so that's the minimum value of $X.$
What is the maximum value of $X$? Can $X = 36pi$?
If you think it can, find a point that could be selected to make $X = 36pi.$
Can $X = 7$?
Find the set of points such that $X leq 5.$ What is the area of that set?
What is the probability that $X leq 5$?
Try this for some other numbers instead of $5.$ Do you see a pattern?
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
$endgroup$
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
add a comment |
$begingroup$
It is true that the uniform random distribution of points inside the circle has a factor of $dfrac{1}{36pi}.$
That is, if someone has in mind a particular subset of points within the circle and that subset of points has area $A,$ then the probability that the randomly selected point will be in that particular subset is $dfrac{A}{36pi}.$
But you have been asked for a random distribution over numbers, and points are not numbers.
What is the minimum value of $X$?
Since $X$ is the distance between the selected point and the center of the circle,
if the selected point is the center of the circle then $X=0.$
You can't have a distance less than $0,$ so that's the minimum value of $X.$
What is the maximum value of $X$? Can $X = 36pi$?
If you think it can, find a point that could be selected to make $X = 36pi.$
Can $X = 7$?
Find the set of points such that $X leq 5.$ What is the area of that set?
What is the probability that $X leq 5$?
Try this for some other numbers instead of $5.$ Do you see a pattern?
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
$endgroup$
It is true that the uniform random distribution of points inside the circle has a factor of $dfrac{1}{36pi}.$
That is, if someone has in mind a particular subset of points within the circle and that subset of points has area $A,$ then the probability that the randomly selected point will be in that particular subset is $dfrac{A}{36pi}.$
But you have been asked for a random distribution over numbers, and points are not numbers.
What is the minimum value of $X$?
Since $X$ is the distance between the selected point and the center of the circle,
if the selected point is the center of the circle then $X=0.$
You can't have a distance less than $0,$ so that's the minimum value of $X.$
What is the maximum value of $X$? Can $X = 36pi$?
If you think it can, find a point that could be selected to make $X = 36pi.$
Can $X = 7$?
Find the set of points such that $X leq 5.$ What is the area of that set?
What is the probability that $X leq 5$?
Try this for some other numbers instead of $5.$ Do you see a pattern?
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
answered Dec 16 '18 at 14:08
David KDavid K
55k344120
55k344120
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
add a comment |
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
Hi thank you for the answer. No I don't see a pattern. Yes it's true that X can have values between 0 and 6 but I don't understand what you are trying to say. Are you saying that we use X to find the constants a and b in the density function?
$endgroup$
– David Daniels
Dec 16 '18 at 16:29
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
$begingroup$
I suspect you don't see the pattern because you have a preconceived notion about the pattern you have to find, and that notion is wrong. What kind of density function has the constants $a$ and $b$ that you mentioned? Whatever it is, I have a hunch it doesn't apply to this question.
$endgroup$
– David K
Dec 16 '18 at 16:39
add a comment |
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4
$begingroup$
Tell me if I'm being dense, but is there anything random ($X$) about the distance of a randomly selected point on the circle from its center? Or are we, in fact, talking about picking a random point from the disc?
$endgroup$
– Matthias
Dec 16 '18 at 1:39