Expectation of almost impossible events The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Zero probability and impossibilityIs it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?Guidelines for Obtaining Recursive Equations?How likely is it not to be anyone's best friend?Conditioning events on a conditional expectationImpossible events that actually happenedAlmost sure bounded imply finite expectation?Ideas for defining a “size” which informally measures subsets of rationals to eachother?Probability on Infinite Sets.Finite expectation induces finite almost surelyI have some unclear things in mind related to integrating a Dirichlet function.
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Expectation of almost impossible events
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Zero probability and impossibilityIs it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?Guidelines for Obtaining Recursive Equations?How likely is it not to be anyone's best friend?Conditioning events on a conditional expectationImpossible events that actually happenedAlmost sure bounded imply finite expectation?Ideas for defining a “size” which informally measures subsets of rationals to eachother?Probability on Infinite Sets.Finite expectation induces finite almost surelyI have some unclear things in mind related to integrating a Dirichlet function.
$begingroup$
In several questions posted here, a satisfying answer, such as 1, may be found on difference between events with zero probability and impossible events. To wrap my mind around this, I usually invoke the interval $I = [0,1]$ as model of explanation - if one picks a countably infinite subset of $I$, with uniformly distributed probability, how many rational numbers are expected in it? Its pointed out to me that zero would be right answer, even some picks may contain rational numbers.
To recollect myself, I'd like to know is that true for any zero-measure subset such as Cantors dust?
probability measure-theory
$endgroup$
|
show 3 more comments
$begingroup$
In several questions posted here, a satisfying answer, such as 1, may be found on difference between events with zero probability and impossible events. To wrap my mind around this, I usually invoke the interval $I = [0,1]$ as model of explanation - if one picks a countably infinite subset of $I$, with uniformly distributed probability, how many rational numbers are expected in it? Its pointed out to me that zero would be right answer, even some picks may contain rational numbers.
To recollect myself, I'd like to know is that true for any zero-measure subset such as Cantors dust?
probability measure-theory
$endgroup$
1
$begingroup$
Not sure what you are asking. Expectation is countably additive (at least for non-negative random variables, like indicator variables), so since each of your sequence has $0$ probability of being rational, the expected number of rational numbers in the sequence is $0$. Beyond that, I can't really follow what you write.
$endgroup$
– lulu
Mar 25 at 10:05
$begingroup$
Yeah, I wrote that really bad - I'll edit. Basically, I didn't mean that the entire sequence is rational, but only finite number of its members to be rational.
$endgroup$
– Stipe Galić
Mar 25 at 10:12
$begingroup$
Also the event that $a_ninmathbb Q$ is expected to appear exactly zero times. In that sense there is no distinction with an impossible event. So your model of explanation is not sound.
$endgroup$
– drhab
Mar 25 at 10:12
$begingroup$
@StipeGalić Once again: the expected number of rational elements in your sequence is $0$. Not "finite". I think you are confused about what "expectation" means.
$endgroup$
– lulu
Mar 25 at 10:14
$begingroup$
The difference between a possible event and an impossible event is that the former can occur and the latter can not. It is perfectly possible that you choose the number $frac 12$ from $[0,1]$, though it has probability $0$. In this way, continuous probability deviates from our intuition (and from basic properties of discrete probability). But in such cases we either need to surrender our intuition or search for a new theory in which our intuition holds up.
$endgroup$
– lulu
Mar 25 at 10:17
|
show 3 more comments
$begingroup$
In several questions posted here, a satisfying answer, such as 1, may be found on difference between events with zero probability and impossible events. To wrap my mind around this, I usually invoke the interval $I = [0,1]$ as model of explanation - if one picks a countably infinite subset of $I$, with uniformly distributed probability, how many rational numbers are expected in it? Its pointed out to me that zero would be right answer, even some picks may contain rational numbers.
To recollect myself, I'd like to know is that true for any zero-measure subset such as Cantors dust?
probability measure-theory
$endgroup$
In several questions posted here, a satisfying answer, such as 1, may be found on difference between events with zero probability and impossible events. To wrap my mind around this, I usually invoke the interval $I = [0,1]$ as model of explanation - if one picks a countably infinite subset of $I$, with uniformly distributed probability, how many rational numbers are expected in it? Its pointed out to me that zero would be right answer, even some picks may contain rational numbers.
To recollect myself, I'd like to know is that true for any zero-measure subset such as Cantors dust?
probability measure-theory
probability measure-theory
edited Mar 25 at 10:25
Stipe Galić
asked Mar 25 at 9:49
Stipe GalićStipe Galić
1143
1143
1
$begingroup$
Not sure what you are asking. Expectation is countably additive (at least for non-negative random variables, like indicator variables), so since each of your sequence has $0$ probability of being rational, the expected number of rational numbers in the sequence is $0$. Beyond that, I can't really follow what you write.
$endgroup$
– lulu
Mar 25 at 10:05
$begingroup$
Yeah, I wrote that really bad - I'll edit. Basically, I didn't mean that the entire sequence is rational, but only finite number of its members to be rational.
$endgroup$
– Stipe Galić
Mar 25 at 10:12
$begingroup$
Also the event that $a_ninmathbb Q$ is expected to appear exactly zero times. In that sense there is no distinction with an impossible event. So your model of explanation is not sound.
$endgroup$
– drhab
Mar 25 at 10:12
$begingroup$
@StipeGalić Once again: the expected number of rational elements in your sequence is $0$. Not "finite". I think you are confused about what "expectation" means.
$endgroup$
– lulu
Mar 25 at 10:14
$begingroup$
The difference between a possible event and an impossible event is that the former can occur and the latter can not. It is perfectly possible that you choose the number $frac 12$ from $[0,1]$, though it has probability $0$. In this way, continuous probability deviates from our intuition (and from basic properties of discrete probability). But in such cases we either need to surrender our intuition or search for a new theory in which our intuition holds up.
$endgroup$
– lulu
Mar 25 at 10:17
|
show 3 more comments
1
$begingroup$
Not sure what you are asking. Expectation is countably additive (at least for non-negative random variables, like indicator variables), so since each of your sequence has $0$ probability of being rational, the expected number of rational numbers in the sequence is $0$. Beyond that, I can't really follow what you write.
$endgroup$
– lulu
Mar 25 at 10:05
$begingroup$
Yeah, I wrote that really bad - I'll edit. Basically, I didn't mean that the entire sequence is rational, but only finite number of its members to be rational.
$endgroup$
– Stipe Galić
Mar 25 at 10:12
$begingroup$
Also the event that $a_ninmathbb Q$ is expected to appear exactly zero times. In that sense there is no distinction with an impossible event. So your model of explanation is not sound.
$endgroup$
– drhab
Mar 25 at 10:12
$begingroup$
@StipeGalić Once again: the expected number of rational elements in your sequence is $0$. Not "finite". I think you are confused about what "expectation" means.
$endgroup$
– lulu
Mar 25 at 10:14
$begingroup$
The difference between a possible event and an impossible event is that the former can occur and the latter can not. It is perfectly possible that you choose the number $frac 12$ from $[0,1]$, though it has probability $0$. In this way, continuous probability deviates from our intuition (and from basic properties of discrete probability). But in such cases we either need to surrender our intuition or search for a new theory in which our intuition holds up.
$endgroup$
– lulu
Mar 25 at 10:17
1
1
$begingroup$
Not sure what you are asking. Expectation is countably additive (at least for non-negative random variables, like indicator variables), so since each of your sequence has $0$ probability of being rational, the expected number of rational numbers in the sequence is $0$. Beyond that, I can't really follow what you write.
$endgroup$
– lulu
Mar 25 at 10:05
$begingroup$
Not sure what you are asking. Expectation is countably additive (at least for non-negative random variables, like indicator variables), so since each of your sequence has $0$ probability of being rational, the expected number of rational numbers in the sequence is $0$. Beyond that, I can't really follow what you write.
$endgroup$
– lulu
Mar 25 at 10:05
$begingroup$
Yeah, I wrote that really bad - I'll edit. Basically, I didn't mean that the entire sequence is rational, but only finite number of its members to be rational.
$endgroup$
– Stipe Galić
Mar 25 at 10:12
$begingroup$
Yeah, I wrote that really bad - I'll edit. Basically, I didn't mean that the entire sequence is rational, but only finite number of its members to be rational.
$endgroup$
– Stipe Galić
Mar 25 at 10:12
$begingroup$
Also the event that $a_ninmathbb Q$ is expected to appear exactly zero times. In that sense there is no distinction with an impossible event. So your model of explanation is not sound.
$endgroup$
– drhab
Mar 25 at 10:12
$begingroup$
Also the event that $a_ninmathbb Q$ is expected to appear exactly zero times. In that sense there is no distinction with an impossible event. So your model of explanation is not sound.
$endgroup$
– drhab
Mar 25 at 10:12
$begingroup$
@StipeGalić Once again: the expected number of rational elements in your sequence is $0$. Not "finite". I think you are confused about what "expectation" means.
$endgroup$
– lulu
Mar 25 at 10:14
$begingroup$
@StipeGalić Once again: the expected number of rational elements in your sequence is $0$. Not "finite". I think you are confused about what "expectation" means.
$endgroup$
– lulu
Mar 25 at 10:14
$begingroup$
The difference between a possible event and an impossible event is that the former can occur and the latter can not. It is perfectly possible that you choose the number $frac 12$ from $[0,1]$, though it has probability $0$. In this way, continuous probability deviates from our intuition (and from basic properties of discrete probability). But in such cases we either need to surrender our intuition or search for a new theory in which our intuition holds up.
$endgroup$
– lulu
Mar 25 at 10:17
$begingroup$
The difference between a possible event and an impossible event is that the former can occur and the latter can not. It is perfectly possible that you choose the number $frac 12$ from $[0,1]$, though it has probability $0$. In this way, continuous probability deviates from our intuition (and from basic properties of discrete probability). But in such cases we either need to surrender our intuition or search for a new theory in which our intuition holds up.
$endgroup$
– lulu
Mar 25 at 10:17
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
I may have grossly misunderstood your question but I think one instance where I felt equally confused initially was with continuous probability density function. Say, you want to pick a random point on the circumference of a circle. In polar co-ordinates, the point can be $(r,theta)$ where $theta$ can be drawn from a uniform distribution in the interval of $[0,2pi]$. In this entire interval, the p.d.f has a value $1/2pi$. So you might think that the probability that $theta$ is $pi/6$ is $1/2pi$. Nope. It is actually zero.
In fact, the probability that any particular value of $theta$ being drawn is zero and yet when a point is chosen it has a definite value. The expectation of $theta$ is a non zero value. So I think that is a good example of zero probability that isn't impossible.
$endgroup$
add a comment |
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$begingroup$
I may have grossly misunderstood your question but I think one instance where I felt equally confused initially was with continuous probability density function. Say, you want to pick a random point on the circumference of a circle. In polar co-ordinates, the point can be $(r,theta)$ where $theta$ can be drawn from a uniform distribution in the interval of $[0,2pi]$. In this entire interval, the p.d.f has a value $1/2pi$. So you might think that the probability that $theta$ is $pi/6$ is $1/2pi$. Nope. It is actually zero.
In fact, the probability that any particular value of $theta$ being drawn is zero and yet when a point is chosen it has a definite value. The expectation of $theta$ is a non zero value. So I think that is a good example of zero probability that isn't impossible.
$endgroup$
add a comment |
$begingroup$
I may have grossly misunderstood your question but I think one instance where I felt equally confused initially was with continuous probability density function. Say, you want to pick a random point on the circumference of a circle. In polar co-ordinates, the point can be $(r,theta)$ where $theta$ can be drawn from a uniform distribution in the interval of $[0,2pi]$. In this entire interval, the p.d.f has a value $1/2pi$. So you might think that the probability that $theta$ is $pi/6$ is $1/2pi$. Nope. It is actually zero.
In fact, the probability that any particular value of $theta$ being drawn is zero and yet when a point is chosen it has a definite value. The expectation of $theta$ is a non zero value. So I think that is a good example of zero probability that isn't impossible.
$endgroup$
add a comment |
$begingroup$
I may have grossly misunderstood your question but I think one instance where I felt equally confused initially was with continuous probability density function. Say, you want to pick a random point on the circumference of a circle. In polar co-ordinates, the point can be $(r,theta)$ where $theta$ can be drawn from a uniform distribution in the interval of $[0,2pi]$. In this entire interval, the p.d.f has a value $1/2pi$. So you might think that the probability that $theta$ is $pi/6$ is $1/2pi$. Nope. It is actually zero.
In fact, the probability that any particular value of $theta$ being drawn is zero and yet when a point is chosen it has a definite value. The expectation of $theta$ is a non zero value. So I think that is a good example of zero probability that isn't impossible.
$endgroup$
I may have grossly misunderstood your question but I think one instance where I felt equally confused initially was with continuous probability density function. Say, you want to pick a random point on the circumference of a circle. In polar co-ordinates, the point can be $(r,theta)$ where $theta$ can be drawn from a uniform distribution in the interval of $[0,2pi]$. In this entire interval, the p.d.f has a value $1/2pi$. So you might think that the probability that $theta$ is $pi/6$ is $1/2pi$. Nope. It is actually zero.
In fact, the probability that any particular value of $theta$ being drawn is zero and yet when a point is chosen it has a definite value. The expectation of $theta$ is a non zero value. So I think that is a good example of zero probability that isn't impossible.
answered Mar 25 at 12:02
Balakrishnan RajanBalakrishnan Rajan
15210
15210
add a comment |
add a comment |
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$begingroup$
Not sure what you are asking. Expectation is countably additive (at least for non-negative random variables, like indicator variables), so since each of your sequence has $0$ probability of being rational, the expected number of rational numbers in the sequence is $0$. Beyond that, I can't really follow what you write.
$endgroup$
– lulu
Mar 25 at 10:05
$begingroup$
Yeah, I wrote that really bad - I'll edit. Basically, I didn't mean that the entire sequence is rational, but only finite number of its members to be rational.
$endgroup$
– Stipe Galić
Mar 25 at 10:12
$begingroup$
Also the event that $a_ninmathbb Q$ is expected to appear exactly zero times. In that sense there is no distinction with an impossible event. So your model of explanation is not sound.
$endgroup$
– drhab
Mar 25 at 10:12
$begingroup$
@StipeGalić Once again: the expected number of rational elements in your sequence is $0$. Not "finite". I think you are confused about what "expectation" means.
$endgroup$
– lulu
Mar 25 at 10:14
$begingroup$
The difference between a possible event and an impossible event is that the former can occur and the latter can not. It is perfectly possible that you choose the number $frac 12$ from $[0,1]$, though it has probability $0$. In this way, continuous probability deviates from our intuition (and from basic properties of discrete probability). But in such cases we either need to surrender our intuition or search for a new theory in which our intuition holds up.
$endgroup$
– lulu
Mar 25 at 10:17