What is the relationship between probability densities and mass densities?
$begingroup$
In statistics we have probability distribution functions which give us likelihood that some random variable ($X$) will equal a particular value. Assuming this variable is continuous, the distribution satisfies:
$$
int f_{X}(x) dx = 1
$$
and can be used to obtain the moments of the variable
begin{align}
mu'_i = int X^i f_x(x) dx
end{align}
In physics we often work with the mass distribution function which describes the distribution of mass. For example the total mass across a system is given by:
$$
M_{tot} = int g_M(m) dm
$$
and like the statistical PDF we can also derive moments of this mass distribution function by integrating across the distribution function:
$$
mu'_i = int m g_M(m) dm
$$
I am wondering what the connection between these two concepts is and if there is any way to move between them?
statistics probability-distributions physics
asked Jan 9 at 17:57
TomTom
436
$endgroup$
add a comment |
$begingroup$
In statistics we have probability distribution functions which give us likelihood that some random variable ($X$) will equal a particular value. Assuming this variable is continuous, the distribution satisfies:
$$
int f_{X}(x) dx = 1
$$
and can be used to obtain the moments of the variable
begin{align}
mu'_i = int X^i f_x(x) dx
end{align}
In physics we often work with the mass distribution function which describes the distribution of mass. For example the total mass across a system is given by:
$$
M_{tot} = int g_M(m) dm
$$
and like the statistical PDF we can also derive moments of this mass distribution function by integrating across the distribution function:
$$
mu'_i = int m g_M(m) dm
$$
I am wondering what the connection between these two concepts is and if there is any way to move between them?
statistics probability-distributions physics
asked Jan 9 at 17:57
TomTom
436
$endgroup$
add a comment |
$begingroup$
In statistics we have probability distribution functions which give us likelihood that some random variable ($X$) will equal a particular value. Assuming this variable is continuous, the distribution satisfies:
$$
int f_{X}(x) dx = 1
$$
and can be used to obtain the moments of the variable
begin{align}
mu'_i = int X^i f_x(x) dx
end{align}
In physics we often work with the mass distribution function which describes the distribution of mass. For example the total mass across a system is given by:
$$
M_{tot} = int g_M(m) dm
$$
and like the statistical PDF we can also derive moments of this mass distribution function by integrating across the distribution function:
$$
mu'_i = int m g_M(m) dm
$$
I am wondering what the connection between these two concepts is and if there is any way to move between them?
statistics probability-distributions physics
asked Jan 9 at 17:57
TomTom
436
$endgroup$
In statistics we have probability distribution functions which give us likelihood that some random variable ($X$) will equal a particular value. Assuming this variable is continuous, the distribution satisfies:
$$
int f_{X}(x) dx = 1
$$
and can be used to obtain the moments of the variable
begin{align}
mu'_i = int X^i f_x(x) dx
end{align}
In physics we often work with the mass distribution function which describes the distribution of mass. For example the total mass across a system is given by:
$$
M_{tot} = int g_M(m) dm
$$
and like the statistical PDF we can also derive moments of this mass distribution function by integrating across the distribution function:
$$
mu'_i = int m g_M(m) dm
$$
I am wondering what the connection between these two concepts is and if there is any way to move between them?
statistics probability-distributions physics
statistics probability-distributions physics
asked Jan 9 at 17:57
TomTom
436
asked Jan 9 at 17:57
TomTom
436
asked Jan 9 at 17:57
TomTom
436
asked Jan 9 at 17:57
TomTom
436
asked Jan 9 at 17:57
TomTom
436
436
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As a very general concept density is the ratio of "stuff" to the "space" in which that "stuff" exists. Mass density is the ratio between mass and volume, population density is the ratio between a population and the area it inhabits, and probability density is the ratio between the probability of an event and the space in which that event occurs.
In the most elementary case, density is uniform, and equal to the total amount of "stuff" divided by the total amount of "space". If the density is not uniform, then you have to use calculus (which gives the same ratio, only in terms of variable quantities). When you integrate the density function, you are finding the total amount of "stuff" (this is equivalent to cancelling out the denominator of the ratio in the elementary case).
You can do this with anything that can be described as "rate-like"; velocity is like the "displacement density" in time, concentration is the "molecular density" in some volume of a substance, etc.
Besides that, there isn't really much of a direct connection. You could make mass density and probability density equivalent by assigning a mass to every event, or by creating a function which gives the probability that the mass at a given point will be equal to some quantity... but I don't see any reason to do that.
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3067765%2fwhat-is-the-relationship-between-probability-densities-and-mass-densities%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As a very general concept density is the ratio of "stuff" to the "space" in which that "stuff" exists. Mass density is the ratio between mass and volume, population density is the ratio between a population and the area it inhabits, and probability density is the ratio between the probability of an event and the space in which that event occurs.
In the most elementary case, density is uniform, and equal to the total amount of "stuff" divided by the total amount of "space". If the density is not uniform, then you have to use calculus (which gives the same ratio, only in terms of variable quantities). When you integrate the density function, you are finding the total amount of "stuff" (this is equivalent to cancelling out the denominator of the ratio in the elementary case).
You can do this with anything that can be described as "rate-like"; velocity is like the "displacement density" in time, concentration is the "molecular density" in some volume of a substance, etc.
Besides that, there isn't really much of a direct connection. You could make mass density and probability density equivalent by assigning a mass to every event, or by creating a function which gives the probability that the mass at a given point will be equal to some quantity... but I don't see any reason to do that.
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
$endgroup$
add a comment |
$begingroup$
As a very general concept density is the ratio of "stuff" to the "space" in which that "stuff" exists. Mass density is the ratio between mass and volume, population density is the ratio between a population and the area it inhabits, and probability density is the ratio between the probability of an event and the space in which that event occurs.
In the most elementary case, density is uniform, and equal to the total amount of "stuff" divided by the total amount of "space". If the density is not uniform, then you have to use calculus (which gives the same ratio, only in terms of variable quantities). When you integrate the density function, you are finding the total amount of "stuff" (this is equivalent to cancelling out the denominator of the ratio in the elementary case).
You can do this with anything that can be described as "rate-like"; velocity is like the "displacement density" in time, concentration is the "molecular density" in some volume of a substance, etc.
Besides that, there isn't really much of a direct connection. You could make mass density and probability density equivalent by assigning a mass to every event, or by creating a function which gives the probability that the mass at a given point will be equal to some quantity... but I don't see any reason to do that.
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
$endgroup$
add a comment |
$begingroup$
As a very general concept density is the ratio of "stuff" to the "space" in which that "stuff" exists. Mass density is the ratio between mass and volume, population density is the ratio between a population and the area it inhabits, and probability density is the ratio between the probability of an event and the space in which that event occurs.
In the most elementary case, density is uniform, and equal to the total amount of "stuff" divided by the total amount of "space". If the density is not uniform, then you have to use calculus (which gives the same ratio, only in terms of variable quantities). When you integrate the density function, you are finding the total amount of "stuff" (this is equivalent to cancelling out the denominator of the ratio in the elementary case).
You can do this with anything that can be described as "rate-like"; velocity is like the "displacement density" in time, concentration is the "molecular density" in some volume of a substance, etc.
Besides that, there isn't really much of a direct connection. You could make mass density and probability density equivalent by assigning a mass to every event, or by creating a function which gives the probability that the mass at a given point will be equal to some quantity... but I don't see any reason to do that.
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
$endgroup$
As a very general concept density is the ratio of "stuff" to the "space" in which that "stuff" exists. Mass density is the ratio between mass and volume, population density is the ratio between a population and the area it inhabits, and probability density is the ratio between the probability of an event and the space in which that event occurs.
In the most elementary case, density is uniform, and equal to the total amount of "stuff" divided by the total amount of "space". If the density is not uniform, then you have to use calculus (which gives the same ratio, only in terms of variable quantities). When you integrate the density function, you are finding the total amount of "stuff" (this is equivalent to cancelling out the denominator of the ratio in the elementary case).
You can do this with anything that can be described as "rate-like"; velocity is like the "displacement density" in time, concentration is the "molecular density" in some volume of a substance, etc.
Besides that, there isn't really much of a direct connection. You could make mass density and probability density equivalent by assigning a mass to every event, or by creating a function which gives the probability that the mass at a given point will be equal to some quantity... but I don't see any reason to do that.
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
answered Jan 9 at 18:32
R. BurtonR. Burton
732110
732110
add a comment |
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.
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%2f3067765%2fwhat-is-the-relationship-between-probability-densities-and-mass-densities%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
