How can I derive the waiting time distribution of gaussian processes?
$begingroup$
Given two successive Gaussian events,
$X_1 sim N(mu,sigma^2),qquad X_2sim N(mu,sigma^2)qquad$ with $X_2 = X_1+epsilon$,$quadepsilonin {rm I!R},quadepsilon>0$
how can I derive the distribution function of the waiting time, $epsilon$, between them?
$X_2 - X_1 = epsilon sim ~?$
In other words, if the time at which a certain event is detected follows a Gaussian distribution, what is, or how can I derive, the distribution of the time interval between two successive events?
[Edit]
My original question was much less detailed than I originally thought, so I will try to give a better example.
Suppose I have a time series with random peaks. If I look at the distribution of those peaks in time I will see a Gaussian distribution of times.
Now I want to look at the distribution of the time interval between peaks (or waiting time). I get this distribution that looks like an exponential distribution. But I know it isn't a true exponential distribution. I want to know which distribution it is or how to derive it.
statistics probability-distributions statistical-mechanics
asked Dec 13 '18 at 4:56
WeißWeiß
63
$endgroup$
|
show 1 more comment
$begingroup$
Given two successive Gaussian events,
$X_1 sim N(mu,sigma^2),qquad X_2sim N(mu,sigma^2)qquad$ with $X_2 = X_1+epsilon$,$quadepsilonin {rm I!R},quadepsilon>0$
how can I derive the distribution function of the waiting time, $epsilon$, between them?
$X_2 - X_1 = epsilon sim ~?$
In other words, if the time at which a certain event is detected follows a Gaussian distribution, what is, or how can I derive, the distribution of the time interval between two successive events?
[Edit]
My original question was much less detailed than I originally thought, so I will try to give a better example.
Suppose I have a time series with random peaks. If I look at the distribution of those peaks in time I will see a Gaussian distribution of times.
Now I want to look at the distribution of the time interval between peaks (or waiting time). I get this distribution that looks like an exponential distribution. But I know it isn't a true exponential distribution. I want to know which distribution it is or how to derive it.
statistics probability-distributions statistical-mechanics
asked Dec 13 '18 at 4:56
WeißWeiß
63
$endgroup$
$begingroup$
Do they share the same mean $mu$ or are the means and variances different?
$endgroup$
– Karn Watcharasupat
Dec 13 '18 at 5:17
$begingroup$
More importantly, what assumptions are you placing on the relationship between $X_2-X_1$ and $X_1$?
$endgroup$
– zoidberg
Dec 13 '18 at 5:35
$begingroup$
A waiting time is a meaningful concept in a Poisson process, but less so with a Gaussian process where the difference can be negative
$endgroup$
– Henry
Dec 13 '18 at 8:13
$begingroup$
They do share the same mean $mu$ and $sigma^2$. About $X_1$ and $X_2$: they are both time values at which a certain event is detected. I know that the distribution of those time values is Gaussian but I want to look at the distribution of the waiting time between two detections.
$endgroup$
– Weiß
Dec 14 '18 at 1:46
$begingroup$
Unfortunately, your addition doesn't really clarify things. You've shown us only summary statistics for the $X_i$ individually. What we need is information about the dependence between the $X_i$.
$endgroup$
– zoidberg
Dec 14 '18 at 2:37
|
show 1 more comment
$begingroup$
Given two successive Gaussian events,
$X_1 sim N(mu,sigma^2),qquad X_2sim N(mu,sigma^2)qquad$ with $X_2 = X_1+epsilon$,$quadepsilonin {rm I!R},quadepsilon>0$
how can I derive the distribution function of the waiting time, $epsilon$, between them?
$X_2 - X_1 = epsilon sim ~?$
In other words, if the time at which a certain event is detected follows a Gaussian distribution, what is, or how can I derive, the distribution of the time interval between two successive events?
[Edit]
My original question was much less detailed than I originally thought, so I will try to give a better example.
Suppose I have a time series with random peaks. If I look at the distribution of those peaks in time I will see a Gaussian distribution of times.
Now I want to look at the distribution of the time interval between peaks (or waiting time). I get this distribution that looks like an exponential distribution. But I know it isn't a true exponential distribution. I want to know which distribution it is or how to derive it.
statistics probability-distributions statistical-mechanics
asked Dec 13 '18 at 4:56
WeißWeiß
63
$endgroup$
Given two successive Gaussian events,
$X_1 sim N(mu,sigma^2),qquad X_2sim N(mu,sigma^2)qquad$ with $X_2 = X_1+epsilon$,$quadepsilonin {rm I!R},quadepsilon>0$
how can I derive the distribution function of the waiting time, $epsilon$, between them?
$X_2 - X_1 = epsilon sim ~?$
In other words, if the time at which a certain event is detected follows a Gaussian distribution, what is, or how can I derive, the distribution of the time interval between two successive events?
[Edit]
My original question was much less detailed than I originally thought, so I will try to give a better example.
Suppose I have a time series with random peaks. If I look at the distribution of those peaks in time I will see a Gaussian distribution of times.
Now I want to look at the distribution of the time interval between peaks (or waiting time). I get this distribution that looks like an exponential distribution. But I know it isn't a true exponential distribution. I want to know which distribution it is or how to derive it.
statistics probability-distributions statistical-mechanics
statistics probability-distributions statistical-mechanics
asked Dec 13 '18 at 4:56
WeißWeiß
63
asked Dec 13 '18 at 4:56
WeißWeiß
63
edited Dec 14 '18 at 2:19
Weiß
asked Dec 13 '18 at 4:56
WeißWeiß
63
asked Dec 13 '18 at 4:56
WeißWeiß
63
asked Dec 13 '18 at 4:56
WeißWeiß
63
63
$begingroup$
Do they share the same mean $mu$ or are the means and variances different?
$endgroup$
– Karn Watcharasupat
Dec 13 '18 at 5:17
$begingroup$
More importantly, what assumptions are you placing on the relationship between $X_2-X_1$ and $X_1$?
$endgroup$
– zoidberg
Dec 13 '18 at 5:35
$begingroup$
A waiting time is a meaningful concept in a Poisson process, but less so with a Gaussian process where the difference can be negative
$endgroup$
– Henry
Dec 13 '18 at 8:13
$begingroup$
They do share the same mean $mu$ and $sigma^2$. About $X_1$ and $X_2$: they are both time values at which a certain event is detected. I know that the distribution of those time values is Gaussian but I want to look at the distribution of the waiting time between two detections.
$endgroup$
– Weiß
Dec 14 '18 at 1:46
$begingroup$
Unfortunately, your addition doesn't really clarify things. You've shown us only summary statistics for the $X_i$ individually. What we need is information about the dependence between the $X_i$.
$endgroup$
– zoidberg
Dec 14 '18 at 2:37
|
show 1 more comment
$begingroup$
Do they share the same mean $mu$ or are the means and variances different?
$endgroup$
– Karn Watcharasupat
Dec 13 '18 at 5:17
$begingroup$
More importantly, what assumptions are you placing on the relationship between $X_2-X_1$ and $X_1$?
$endgroup$
– zoidberg
Dec 13 '18 at 5:35
$begingroup$
A waiting time is a meaningful concept in a Poisson process, but less so with a Gaussian process where the difference can be negative
$endgroup$
– Henry
Dec 13 '18 at 8:13
$begingroup$
They do share the same mean $mu$ and $sigma^2$. About $X_1$ and $X_2$: they are both time values at which a certain event is detected. I know that the distribution of those time values is Gaussian but I want to look at the distribution of the waiting time between two detections.
$endgroup$
– Weiß
Dec 14 '18 at 1:46
$begingroup$
Unfortunately, your addition doesn't really clarify things. You've shown us only summary statistics for the $X_i$ individually. What we need is information about the dependence between the $X_i$.
$endgroup$
– zoidberg
Dec 14 '18 at 2:37
$begingroup$
Do they share the same mean $mu$ or are the means and variances different?
$endgroup$
– Karn Watcharasupat
Dec 13 '18 at 5:17
$begingroup$
Do they share the same mean $mu$ or are the means and variances different?
$endgroup$
– Karn Watcharasupat
Dec 13 '18 at 5:17
$begingroup$
More importantly, what assumptions are you placing on the relationship between $X_2-X_1$ and $X_1$?
$endgroup$
– zoidberg
Dec 13 '18 at 5:35
$begingroup$
More importantly, what assumptions are you placing on the relationship between $X_2-X_1$ and $X_1$?
$endgroup$
– zoidberg
Dec 13 '18 at 5:35
$begingroup$
A waiting time is a meaningful concept in a Poisson process, but less so with a Gaussian process where the difference can be negative
$endgroup$
– Henry
Dec 13 '18 at 8:13
$begingroup$
A waiting time is a meaningful concept in a Poisson process, but less so with a Gaussian process where the difference can be negative
$endgroup$
– Henry
Dec 13 '18 at 8:13
$begingroup$
They do share the same mean $mu$ and $sigma^2$. About $X_1$ and $X_2$: they are both time values at which a certain event is detected. I know that the distribution of those time values is Gaussian but I want to look at the distribution of the waiting time between two detections.
$endgroup$
– Weiß
Dec 14 '18 at 1:46
$begingroup$
They do share the same mean $mu$ and $sigma^2$. About $X_1$ and $X_2$: they are both time values at which a certain event is detected. I know that the distribution of those time values is Gaussian but I want to look at the distribution of the waiting time between two detections.
$endgroup$
– Weiß
Dec 14 '18 at 1:46
$begingroup$
Unfortunately, your addition doesn't really clarify things. You've shown us only summary statistics for the $X_i$ individually. What we need is information about the dependence between the $X_i$.
$endgroup$
– zoidberg
Dec 14 '18 at 2:37
$begingroup$
Unfortunately, your addition doesn't really clarify things. You've shown us only summary statistics for the $X_i$ individually. What we need is information about the dependence between the $X_i$.
$endgroup$
– zoidberg
Dec 14 '18 at 2:37
|
show 1 more comment
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$begingroup$
Do they share the same mean $mu$ or are the means and variances different?
$endgroup$
– Karn Watcharasupat
Dec 13 '18 at 5:17
$begingroup$
More importantly, what assumptions are you placing on the relationship between $X_2-X_1$ and $X_1$?
$endgroup$
– zoidberg
Dec 13 '18 at 5:35
$begingroup$
A waiting time is a meaningful concept in a Poisson process, but less so with a Gaussian process where the difference can be negative
$endgroup$
– Henry
Dec 13 '18 at 8:13
$begingroup$
They do share the same mean $mu$ and $sigma^2$. About $X_1$ and $X_2$: they are both time values at which a certain event is detected. I know that the distribution of those time values is Gaussian but I want to look at the distribution of the waiting time between two detections.
$endgroup$
– Weiß
Dec 14 '18 at 1:46
$begingroup$
Unfortunately, your addition doesn't really clarify things. You've shown us only summary statistics for the $X_i$ individually. What we need is information about the dependence between the $X_i$.
$endgroup$
– zoidberg
Dec 14 '18 at 2:37