Probability distribution vs. probability mass function / Probability density function terms: what's the...
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I struggle to understand what's the difference between Probability distribution (https://en.wikipedia.org/wiki/Probability_distribution) vs. probability mass function (https://en.wikipedia.org/wiki/Probability_mass_function) or Probability density function (https://en.wikipedia.org/wiki/Probability_density_function). Both probability distribution and lets say PMF seem to reflect probability of values of a random variable. Note that I do not ask the difference between PDF and PMF.
Consider the following example when a 4-sides dice is rolled twice. X is the sum of two throws. I calculate the probability mass function (left) and then show the result graphically (right). But it seems that it is fair to call this graph probability distribution. Isn't it?
Thanks!
probability probability-theory probability-distributions
asked Dec 3 '18 at 18:32
JohnJohn
1236
$endgroup$
|
show 4 more comments
$begingroup$
I struggle to understand what's the difference between Probability distribution (https://en.wikipedia.org/wiki/Probability_distribution) vs. probability mass function (https://en.wikipedia.org/wiki/Probability_mass_function) or Probability density function (https://en.wikipedia.org/wiki/Probability_density_function). Both probability distribution and lets say PMF seem to reflect probability of values of a random variable. Note that I do not ask the difference between PDF and PMF.
Consider the following example when a 4-sides dice is rolled twice. X is the sum of two throws. I calculate the probability mass function (left) and then show the result graphically (right). But it seems that it is fair to call this graph probability distribution. Isn't it?
Thanks!
probability probability-theory probability-distributions
asked Dec 3 '18 at 18:32
JohnJohn
1236
$endgroup$
1
$begingroup$
The key is that the Probability Mass Function is associated to discrete random variables, while the Probability Distribution Function is associated to continuous random variables. There are some authors that use this while others simply call it PDF. While the PMF is defined as $P(X=x)$ the PDF could be heuristically defined as $P(x+epsilon geq X geq x-epsilon)$, since for a continuous r.v. $P(X=x)=0$
$endgroup$
– Ramiro Scorolli
Dec 3 '18 at 18:45
$begingroup$
In the context of measure theory a probability distribution is commonly a pushforward-measure induced by a random variable/vector. This is the Kolmogorov-definition and the measure is prescribed by $Bmapsto P(Xin B)$. Using the Kolmogorov-definition we do not have to distinguish between concepts like discrete distributions (linked to a PMF), absolutely continuous distributions (linked to a PDF) or something in between.
$endgroup$
– drhab
Dec 3 '18 at 18:53
$begingroup$
Thanks, but I am asking not about "Probability Distribution Function", but about "Probability distribution" (en.wikipedia.org/wiki/Probability_distribution). Probability distribution can be binomial which is discrete.
$endgroup$
– John
Dec 3 '18 at 18:54
$begingroup$
My former comment concerns "probability distribution" and the Kolmogorov-definition can be found on the link you provide.
$endgroup$
– drhab
Dec 3 '18 at 18:58
$begingroup$
My response was for the first answer :)
$endgroup$
– John
Dec 3 '18 at 19:01
|
show 4 more comments
$begingroup$
I struggle to understand what's the difference between Probability distribution (https://en.wikipedia.org/wiki/Probability_distribution) vs. probability mass function (https://en.wikipedia.org/wiki/Probability_mass_function) or Probability density function (https://en.wikipedia.org/wiki/Probability_density_function). Both probability distribution and lets say PMF seem to reflect probability of values of a random variable. Note that I do not ask the difference between PDF and PMF.
Consider the following example when a 4-sides dice is rolled twice. X is the sum of two throws. I calculate the probability mass function (left) and then show the result graphically (right). But it seems that it is fair to call this graph probability distribution. Isn't it?
Thanks!
probability probability-theory probability-distributions
asked Dec 3 '18 at 18:32
JohnJohn
1236
$endgroup$
I struggle to understand what's the difference between Probability distribution (https://en.wikipedia.org/wiki/Probability_distribution) vs. probability mass function (https://en.wikipedia.org/wiki/Probability_mass_function) or Probability density function (https://en.wikipedia.org/wiki/Probability_density_function). Both probability distribution and lets say PMF seem to reflect probability of values of a random variable. Note that I do not ask the difference between PDF and PMF.
Consider the following example when a 4-sides dice is rolled twice. X is the sum of two throws. I calculate the probability mass function (left) and then show the result graphically (right). But it seems that it is fair to call this graph probability distribution. Isn't it?
Thanks!
probability probability-theory probability-distributions
probability probability-theory probability-distributions
asked Dec 3 '18 at 18:32
JohnJohn
1236
asked Dec 3 '18 at 18:32
JohnJohn
1236
edited Dec 3 '18 at 19:07
John
asked Dec 3 '18 at 18:32
JohnJohn
1236
asked Dec 3 '18 at 18:32
JohnJohn
1236
asked Dec 3 '18 at 18:32
JohnJohn
1236
1236
1
$begingroup$
The key is that the Probability Mass Function is associated to discrete random variables, while the Probability Distribution Function is associated to continuous random variables. There are some authors that use this while others simply call it PDF. While the PMF is defined as $P(X=x)$ the PDF could be heuristically defined as $P(x+epsilon geq X geq x-epsilon)$, since for a continuous r.v. $P(X=x)=0$
$endgroup$
– Ramiro Scorolli
Dec 3 '18 at 18:45
$begingroup$
In the context of measure theory a probability distribution is commonly a pushforward-measure induced by a random variable/vector. This is the Kolmogorov-definition and the measure is prescribed by $Bmapsto P(Xin B)$. Using the Kolmogorov-definition we do not have to distinguish between concepts like discrete distributions (linked to a PMF), absolutely continuous distributions (linked to a PDF) or something in between.
$endgroup$
– drhab
Dec 3 '18 at 18:53
$begingroup$
Thanks, but I am asking not about "Probability Distribution Function", but about "Probability distribution" (en.wikipedia.org/wiki/Probability_distribution). Probability distribution can be binomial which is discrete.
$endgroup$
– John
Dec 3 '18 at 18:54
$begingroup$
My former comment concerns "probability distribution" and the Kolmogorov-definition can be found on the link you provide.
$endgroup$
– drhab
Dec 3 '18 at 18:58
$begingroup$
My response was for the first answer :)
$endgroup$
– John
Dec 3 '18 at 19:01
|
show 4 more comments
1
$begingroup$
The key is that the Probability Mass Function is associated to discrete random variables, while the Probability Distribution Function is associated to continuous random variables. There are some authors that use this while others simply call it PDF. While the PMF is defined as $P(X=x)$ the PDF could be heuristically defined as $P(x+epsilon geq X geq x-epsilon)$, since for a continuous r.v. $P(X=x)=0$
$endgroup$
– Ramiro Scorolli
Dec 3 '18 at 18:45
$begingroup$
In the context of measure theory a probability distribution is commonly a pushforward-measure induced by a random variable/vector. This is the Kolmogorov-definition and the measure is prescribed by $Bmapsto P(Xin B)$. Using the Kolmogorov-definition we do not have to distinguish between concepts like discrete distributions (linked to a PMF), absolutely continuous distributions (linked to a PDF) or something in between.
$endgroup$
– drhab
Dec 3 '18 at 18:53
$begingroup$
Thanks, but I am asking not about "Probability Distribution Function", but about "Probability distribution" (en.wikipedia.org/wiki/Probability_distribution). Probability distribution can be binomial which is discrete.
$endgroup$
– John
Dec 3 '18 at 18:54
$begingroup$
My former comment concerns "probability distribution" and the Kolmogorov-definition can be found on the link you provide.
$endgroup$
– drhab
Dec 3 '18 at 18:58
$begingroup$
My response was for the first answer :)
$endgroup$
– John
Dec 3 '18 at 19:01
1
1
$begingroup$
The key is that the Probability Mass Function is associated to discrete random variables, while the Probability Distribution Function is associated to continuous random variables. There are some authors that use this while others simply call it PDF. While the PMF is defined as $P(X=x)$ the PDF could be heuristically defined as $P(x+epsilon geq X geq x-epsilon)$, since for a continuous r.v. $P(X=x)=0$
$endgroup$
– Ramiro Scorolli
Dec 3 '18 at 18:45
$begingroup$
The key is that the Probability Mass Function is associated to discrete random variables, while the Probability Distribution Function is associated to continuous random variables. There are some authors that use this while others simply call it PDF. While the PMF is defined as $P(X=x)$ the PDF could be heuristically defined as $P(x+epsilon geq X geq x-epsilon)$, since for a continuous r.v. $P(X=x)=0$
$endgroup$
– Ramiro Scorolli
Dec 3 '18 at 18:45
$begingroup$
In the context of measure theory a probability distribution is commonly a pushforward-measure induced by a random variable/vector. This is the Kolmogorov-definition and the measure is prescribed by $Bmapsto P(Xin B)$. Using the Kolmogorov-definition we do not have to distinguish between concepts like discrete distributions (linked to a PMF), absolutely continuous distributions (linked to a PDF) or something in between.
$endgroup$
– drhab
Dec 3 '18 at 18:53
$begingroup$
In the context of measure theory a probability distribution is commonly a pushforward-measure induced by a random variable/vector. This is the Kolmogorov-definition and the measure is prescribed by $Bmapsto P(Xin B)$. Using the Kolmogorov-definition we do not have to distinguish between concepts like discrete distributions (linked to a PMF), absolutely continuous distributions (linked to a PDF) or something in between.
$endgroup$
– drhab
Dec 3 '18 at 18:53
$begingroup$
Thanks, but I am asking not about "Probability Distribution Function", but about "Probability distribution" (en.wikipedia.org/wiki/Probability_distribution). Probability distribution can be binomial which is discrete.
$endgroup$
– John
Dec 3 '18 at 18:54
$begingroup$
Thanks, but I am asking not about "Probability Distribution Function", but about "Probability distribution" (en.wikipedia.org/wiki/Probability_distribution). Probability distribution can be binomial which is discrete.
$endgroup$
– John
Dec 3 '18 at 18:54
$begingroup$
My former comment concerns "probability distribution" and the Kolmogorov-definition can be found on the link you provide.
$endgroup$
– drhab
Dec 3 '18 at 18:58
$begingroup$
My former comment concerns "probability distribution" and the Kolmogorov-definition can be found on the link you provide.
$endgroup$
– drhab
Dec 3 '18 at 18:58
$begingroup$
My response was for the first answer :)
$endgroup$
– John
Dec 3 '18 at 19:01
$begingroup$
My response was for the first answer :)
$endgroup$
– John
Dec 3 '18 at 19:01
|
show 4 more comments
2 Answers
2
active
oldest
votes
$begingroup$
"Probability distribution" is a general term describing a mathematical entity that is represented by the "cumulative distribution function" (or just "distribution function") and also by its "probability mass function" or "probability density function" (or just "density"), when it exists.
For example the following sentence is perfectly correct even though a bit wordy: "the cumulative distribution function of the Normal probability distribution is XXX, while its probability density function is YYY".
As to what your graph reflects, the cumulative distribution function is non-decreasing by definition.
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
$endgroup$
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
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@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
add a comment |
$begingroup$
Probability density functions are always associated with continuous random variables. Continuous variables, are variables which can be measured, such as time, and length. However, a probability mass function is a function which only takes in discrete values for its random variable. However; in both cases the function must satisfy two conditions in order to be a PDF or PMF: 1) The honesty condition (The sum of all the values or outcomes must equal one for discrete cases, and integral for continuous cases). 2) Given any outcome, x, the function f(x) must be between 0 and 1. A probability distribution is a function which assigns a probability to an outcome.
$endgroup$
$begingroup$
Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
$endgroup$
– John
Dec 3 '18 at 19:52
$begingroup$
Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
$endgroup$
– John
Dec 3 '18 at 19:57
$begingroup$
The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
$endgroup$
– drhab
Dec 4 '18 at 7:35
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
"Probability distribution" is a general term describing a mathematical entity that is represented by the "cumulative distribution function" (or just "distribution function") and also by its "probability mass function" or "probability density function" (or just "density"), when it exists.
For example the following sentence is perfectly correct even though a bit wordy: "the cumulative distribution function of the Normal probability distribution is XXX, while its probability density function is YYY".
As to what your graph reflects, the cumulative distribution function is non-decreasing by definition.
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
$endgroup$
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
$begingroup$
@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
add a comment |
$begingroup$
"Probability distribution" is a general term describing a mathematical entity that is represented by the "cumulative distribution function" (or just "distribution function") and also by its "probability mass function" or "probability density function" (or just "density"), when it exists.
For example the following sentence is perfectly correct even though a bit wordy: "the cumulative distribution function of the Normal probability distribution is XXX, while its probability density function is YYY".
As to what your graph reflects, the cumulative distribution function is non-decreasing by definition.
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
$endgroup$
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
$begingroup$
@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
add a comment |
$begingroup$
"Probability distribution" is a general term describing a mathematical entity that is represented by the "cumulative distribution function" (or just "distribution function") and also by its "probability mass function" or "probability density function" (or just "density"), when it exists.
For example the following sentence is perfectly correct even though a bit wordy: "the cumulative distribution function of the Normal probability distribution is XXX, while its probability density function is YYY".
As to what your graph reflects, the cumulative distribution function is non-decreasing by definition.
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
$endgroup$
"Probability distribution" is a general term describing a mathematical entity that is represented by the "cumulative distribution function" (or just "distribution function") and also by its "probability mass function" or "probability density function" (or just "density"), when it exists.
For example the following sentence is perfectly correct even though a bit wordy: "the cumulative distribution function of the Normal probability distribution is XXX, while its probability density function is YYY".
As to what your graph reflects, the cumulative distribution function is non-decreasing by definition.
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
answered Dec 4 '18 at 3:56
Alecos PapadopoulosAlecos Papadopoulos
8,23811535
8,23811535
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
$begingroup$
@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
add a comment |
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
$begingroup$
@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
Thank you a lot! You answer was very helpful.
$endgroup$
– John
Dec 4 '18 at 14:46
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
$begingroup$
But does not CDF should not look like this? probabilitycourse.com/chapter3/3_2_1_cdf.php (Fig. 3.4)
$endgroup$
– John
Dec 4 '18 at 16:09
$begingroup$
@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
@John. Indeed, for a discrete random variable. Where is the problem?
$endgroup$
– Alecos Papadopoulos
Dec 4 '18 at 20:47
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
But in your answer you said that my graph (in the original question) reflects CDF. But in my original graph there is no cumulative aspect, in my view....
$endgroup$
– John
Dec 5 '18 at 8:32
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
$begingroup$
@John I did no such thing. Perhaps one more careful reading of my answrer would be in order.
$endgroup$
– Alecos Papadopoulos
Dec 5 '18 at 9:12
add a comment |
$begingroup$
Probability density functions are always associated with continuous random variables. Continuous variables, are variables which can be measured, such as time, and length. However, a probability mass function is a function which only takes in discrete values for its random variable. However; in both cases the function must satisfy two conditions in order to be a PDF or PMF: 1) The honesty condition (The sum of all the values or outcomes must equal one for discrete cases, and integral for continuous cases). 2) Given any outcome, x, the function f(x) must be between 0 and 1. A probability distribution is a function which assigns a probability to an outcome.
$endgroup$
$begingroup$
Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
$endgroup$
– John
Dec 3 '18 at 19:52
$begingroup$
Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
$endgroup$
– John
Dec 3 '18 at 19:57
$begingroup$
The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
$endgroup$
– drhab
Dec 4 '18 at 7:35
add a comment |
$begingroup$
Probability density functions are always associated with continuous random variables. Continuous variables, are variables which can be measured, such as time, and length. However, a probability mass function is a function which only takes in discrete values for its random variable. However; in both cases the function must satisfy two conditions in order to be a PDF or PMF: 1) The honesty condition (The sum of all the values or outcomes must equal one for discrete cases, and integral for continuous cases). 2) Given any outcome, x, the function f(x) must be between 0 and 1. A probability distribution is a function which assigns a probability to an outcome.
$endgroup$
$begingroup$
Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
$endgroup$
– John
Dec 3 '18 at 19:52
$begingroup$
Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
$endgroup$
– John
Dec 3 '18 at 19:57
$begingroup$
The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
$endgroup$
– drhab
Dec 4 '18 at 7:35
add a comment |
$begingroup$
Probability density functions are always associated with continuous random variables. Continuous variables, are variables which can be measured, such as time, and length. However, a probability mass function is a function which only takes in discrete values for its random variable. However; in both cases the function must satisfy two conditions in order to be a PDF or PMF: 1) The honesty condition (The sum of all the values or outcomes must equal one for discrete cases, and integral for continuous cases). 2) Given any outcome, x, the function f(x) must be between 0 and 1. A probability distribution is a function which assigns a probability to an outcome.
$endgroup$
Probability density functions are always associated with continuous random variables. Continuous variables, are variables which can be measured, such as time, and length. However, a probability mass function is a function which only takes in discrete values for its random variable. However; in both cases the function must satisfy two conditions in order to be a PDF or PMF: 1) The honesty condition (The sum of all the values or outcomes must equal one for discrete cases, and integral for continuous cases). 2) Given any outcome, x, the function f(x) must be between 0 and 1. A probability distribution is a function which assigns a probability to an outcome.
edited Dec 3 '18 at 19:34
answered Dec 3 '18 at 19:29
user612135
$begingroup$
Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
$endgroup$
– John
Dec 3 '18 at 19:52
$begingroup$
Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
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– John
Dec 3 '18 at 19:57
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The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
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– drhab
Dec 4 '18 at 7:35
add a comment |
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Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
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– John
Dec 3 '18 at 19:52
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Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
$endgroup$
– John
Dec 3 '18 at 19:57
$begingroup$
The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
$endgroup$
– drhab
Dec 4 '18 at 7:35
$begingroup$
Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
$endgroup$
– John
Dec 3 '18 at 19:52
$begingroup$
Thank you. In my figure, did I plot: a) probability distribution; b) PMF c)both ?
$endgroup$
– John
Dec 3 '18 at 19:52
$begingroup$
Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
$endgroup$
– John
Dec 3 '18 at 19:57
$begingroup$
Here it is written that "PMF p(S) specifies the probability distribution for the sum S of counts from two dice." Will you agree? en.wikipedia.org/wiki/Probability_distribution#/media/…
$endgroup$
– John
Dec 3 '18 at 19:57
$begingroup$
The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
$endgroup$
– drhab
Dec 4 '18 at 7:35
$begingroup$
The second condition must be satisfied by a PMF but not necessarily by a PDF. The values taken by a PDF are not probabilities and can exceed $1$.
$endgroup$
– drhab
Dec 4 '18 at 7:35
add a comment |
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The key is that the Probability Mass Function is associated to discrete random variables, while the Probability Distribution Function is associated to continuous random variables. There are some authors that use this while others simply call it PDF. While the PMF is defined as $P(X=x)$ the PDF could be heuristically defined as $P(x+epsilon geq X geq x-epsilon)$, since for a continuous r.v. $P(X=x)=0$
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– Ramiro Scorolli
Dec 3 '18 at 18:45
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In the context of measure theory a probability distribution is commonly a pushforward-measure induced by a random variable/vector. This is the Kolmogorov-definition and the measure is prescribed by $Bmapsto P(Xin B)$. Using the Kolmogorov-definition we do not have to distinguish between concepts like discrete distributions (linked to a PMF), absolutely continuous distributions (linked to a PDF) or something in between.
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– drhab
Dec 3 '18 at 18:53
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Thanks, but I am asking not about "Probability Distribution Function", but about "Probability distribution" (en.wikipedia.org/wiki/Probability_distribution). Probability distribution can be binomial which is discrete.
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– John
Dec 3 '18 at 18:54
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My former comment concerns "probability distribution" and the Kolmogorov-definition can be found on the link you provide.
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– drhab
Dec 3 '18 at 18:58
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My response was for the first answer :)
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– John
Dec 3 '18 at 19:01