Difference between “probability density function” and “probability distribution function”? The 2019 Stack Overflow Developer Survey Results Are InDifference between density and distribution [in formal mathematical terms]Probability distribution vs. probability mass function (PMF): what is the difference between the terms?Distinguish Normal Distribution, Gaussian Distribution and Normalised Gaussian Distribution?Confusion between probability distribution function and probability density functionProbability density/mass functionProbability mass function and Probability density functionDifference between Probability and Probability Densityprobability density function and cumulative distribution function.Probability measure, probability density function or probability event ? Are they different?Difference Between joint probability distribution and conditional probability distribution?What is the difference between a probability mass function and discrete probability distribution?What is the difference between a joint distribution function and the joint density function.Probability distribution vs. probability mass function / Probability density function terms: what's the difference
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Difference between “probability density function” and “probability distribution function”?
The 2019 Stack Overflow Developer Survey Results Are InDifference between density and distribution [in formal mathematical terms]Probability distribution vs. probability mass function (PMF): what is the difference between the terms?Distinguish Normal Distribution, Gaussian Distribution and Normalised Gaussian Distribution?Confusion between probability distribution function and probability density functionProbability density/mass functionProbability mass function and Probability density functionDifference between Probability and Probability Densityprobability density function and cumulative distribution function.Probability measure, probability density function or probability event ? Are they different?Difference Between joint probability distribution and conditional probability distribution?What is the difference between a probability mass function and discrete probability distribution?What is the difference between a joint distribution function and the joint density function.Probability distribution vs. probability mass function / Probability density function terms: what's the difference
31
$begingroup$
I am studying for my statistics exam, and have to know a lot of theory. My question is:
Whats the difference between probability density function and probability distribution function?
probability probability-distributions terminology
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
$endgroup$
add a comment |
31
$begingroup$
I am studying for my statistics exam, and have to know a lot of theory. My question is:
Whats the difference between probability density function and probability distribution function?
probability probability-distributions terminology
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
$endgroup$
2
$begingroup$
The density (when it exists) is the derivative of the distribution function.
$endgroup$
– Joel Cohen
Jul 27 '12 at 13:31
1
$begingroup$
You mean, "Difference between Probability density function and cumulative distribution function?"?
$endgroup$
– Matt O'Brien
Feb 5 '14 at 21:08
add a comment |
31
31
31
20
$begingroup$
I am studying for my statistics exam, and have to know a lot of theory. My question is:
Whats the difference between probability density function and probability distribution function?
probability probability-distributions terminology
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
$endgroup$
I am studying for my statistics exam, and have to know a lot of theory. My question is:
Whats the difference between probability density function and probability distribution function?
probability probability-distributions terminology
probability probability-distributions terminology
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
edited Oct 14 '15 at 4:56
user147263
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
asked Jul 27 '12 at 13:28
Le ChifreLe Chifre
5923821
5923821
2
$begingroup$
The density (when it exists) is the derivative of the distribution function.
$endgroup$
– Joel Cohen
Jul 27 '12 at 13:31
1
$begingroup$
You mean, "Difference between Probability density function and cumulative distribution function?"?
$endgroup$
– Matt O'Brien
Feb 5 '14 at 21:08
add a comment |
2
$begingroup$
The density (when it exists) is the derivative of the distribution function.
$endgroup$
– Joel Cohen
Jul 27 '12 at 13:31
1
$begingroup$
You mean, "Difference between Probability density function and cumulative distribution function?"?
$endgroup$
– Matt O'Brien
Feb 5 '14 at 21:08
2
2
$begingroup$
The density (when it exists) is the derivative of the distribution function.
$endgroup$
– Joel Cohen
Jul 27 '12 at 13:31
$begingroup$
The density (when it exists) is the derivative of the distribution function.
$endgroup$
– Joel Cohen
Jul 27 '12 at 13:31
1
1
$begingroup$
You mean, "Difference between Probability density function and cumulative distribution function?"?
$endgroup$
– Matt O'Brien
Feb 5 '14 at 21:08
$begingroup$
You mean, "Difference between Probability density function and cumulative distribution function?"?
$endgroup$
– Matt O'Brien
Feb 5 '14 at 21:08
add a comment |
3 Answers
3
active
oldest
votes
10
$begingroup$
The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is
$$
F(k) = sum_i le k f(i)
$$
if $f$ is discrete and
$$
F(x) = int_y le x f(y),dy
$$
if $f$ is continuous.
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
$endgroup$
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
add a comment |
32
$begingroup$
Distribution Function
- The probability distribution function / probability function has ambiguous definition. They may be referred to:
- Probability density function (PDF)
- Cumulative distribution function (CDF)
- or probability mass function (PMF) (statement from Wikipedia)
- But what confirm is:
- Discrete case: Probability Mass Function (PMF)
- Continuous case: Probability Density Function (PDF)
- Both cases: Cumulative distribution function (CDF)
- Probability at certain $x$ value, $P(X = x)$ can be directly obtained in:
- PMF for discrete case
- PDF for continuous case
- Probability for values less than $x$, $P(X < x)$ or Probability for values within a range from $a$ to $b$, $P(a < X < b)$ can be directly obtained in:
- CDF for both discrete / continuous case
- Distribution function is referred to CDF or Cumulative Frequency Function (see this)
In terms of Acquisition and Plot Generation Method
- Collected data appear as discrete when:
- The measurement of a subject is naturally discrete type, such as numbers resulted from dice rolled, count of people.
- The measurement is digitized machine data, which has no intermediate values between quantized levels due to sampling process.
- In later case, when resolution higher, the measurement is closer to analog/continuous signal/variable.
- Way of generate a PMF from discrete data:
- Plot a histogram of the data for all the $x$'s, the $y$-axis is the frequency or quantity at every $x$.
- Scale the $y$-axis by dividing with total number of data collected (data size) $longrightarrow$ and this is called PMF.
- Way of generate a PDF from discrete / continuous data:
- Find a continuous equation that models the collected data, let say normal distribution equation.
- Calculate the parameters required in the equation from the collected data. For example, parameters for normal distribution equation are mean and standard deviation. Calculate them from collected data.
- Based on the parameters, plot the equation with continuous $x$-value $longrightarrow$ that is called PDF.
- How to generate a CDF:
- In discrete case, CDF accumulates the $y$ values in PMF at each discrete $x$ and less than $x$. Repeat this for every $x$. The final plot is a monotonically increasing until $1$ in the last $x$ $longrightarrow$ this is called discrete CDF.
- In continuous case, integrate PDF over $x$; the result is a continuous CDF.
Why PMF, PDF and CDF?
- PMF is preferred when
- Probability at every $x$ value is interest of study. This makes sense when studying a discrete data - such as we interest to probability of getting certain number from a dice roll.
- PDF is preferred when
- We wish to model a collected data with a continuous function, by using few parameters such as mean to speculate the population distribution.
- CDF is preferred when
- Cumulative probability in a range is point of interest.
- Especially in the case of continuous data, CDF much makes sense than PDF - e.g., probability of students' height less than $170$ cm (CDF) is much informative than the probability at exact $170$ cm (PDF).
edited Mar 24 at 2:42
Rócherz
3,0263823
answered Mar 3 '14 at 10:14
user132704user132704
321133
$endgroup$
add a comment |
0
$begingroup$
Some abuse of language exists in these terms, which can vary. Below is a common usage.
In the continuous case (density):
(continuous) probability distribution function = probability density function = density function
(continuous) probability distribution = density
In the discrete case (mass/distribution):
(discrete) probability distribution function = probability mass function
(discrete) probability distribution = distribution
Oddly enough, you may never see a probability mass function called a mass function or a distribution function, nor may you see a discrete probability distribution called a mass. I am sure there is some historical reason as to why. As they say, das war schon immer so und wird auch immer so bleiben.
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
$endgroup$
add a comment |
protected by Zev Chonoles Aug 23 '16 at 9:24
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
10
$begingroup$
The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is
$$
F(k) = sum_i le k f(i)
$$
if $f$ is discrete and
$$
F(x) = int_y le x f(y),dy
$$
if $f$ is continuous.
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
$endgroup$
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
add a comment |
10
$begingroup$
The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is
$$
F(k) = sum_i le k f(i)
$$
if $f$ is discrete and
$$
F(x) = int_y le x f(y),dy
$$
if $f$ is continuous.
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
$endgroup$
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
add a comment |
10
10
10
$begingroup$
The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is
$$
F(k) = sum_i le k f(i)
$$
if $f$ is discrete and
$$
F(x) = int_y le x f(y),dy
$$
if $f$ is continuous.
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
$endgroup$
The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is
$$
F(k) = sum_i le k f(i)
$$
if $f$ is discrete and
$$
F(x) = int_y le x f(y),dy
$$
if $f$ is continuous.
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
answered Jul 27 '12 at 13:39
August KarlstromAugust Karlstrom
26726
26726
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
add a comment |
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
what is meant by discrete and continuous?
$endgroup$
– Le Chifre
Jul 27 '12 at 13:40
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus if the variable ranges over a discrete or continuous set of values. So if you're rolling a die, you have $1,2,3,4,5,6$, which is discrete. If you're picking a random point on a line, then your set is, say, the interval $[0,L]$ which is continuous.
$endgroup$
– Robert Mastragostino
Jul 27 '12 at 13:45
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
@maximus For example, when flipping a coin or rolling a dice the outcome is discrete whereas measuring the time until the bus arrives at a bus stop is continuous.
$endgroup$
– August Karlstrom
Jul 27 '12 at 13:47
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
so discrete is when you can count it! and continuous is when there is much more probability in it? Is this description right or wrong? Pls correct me!
$endgroup$
– Le Chifre
Jul 27 '12 at 13:50
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
$begingroup$
@maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition.
$endgroup$
– August Karlstrom
Jul 27 '12 at 14:12
add a comment |
32
$begingroup$
Distribution Function
- The probability distribution function / probability function has ambiguous definition. They may be referred to:
- Probability density function (PDF)
- Cumulative distribution function (CDF)
- or probability mass function (PMF) (statement from Wikipedia)
- But what confirm is:
- Discrete case: Probability Mass Function (PMF)
- Continuous case: Probability Density Function (PDF)
- Both cases: Cumulative distribution function (CDF)
- Probability at certain $x$ value, $P(X = x)$ can be directly obtained in:
- PMF for discrete case
- PDF for continuous case
- Probability for values less than $x$, $P(X < x)$ or Probability for values within a range from $a$ to $b$, $P(a < X < b)$ can be directly obtained in:
- CDF for both discrete / continuous case
- Distribution function is referred to CDF or Cumulative Frequency Function (see this)
In terms of Acquisition and Plot Generation Method
- Collected data appear as discrete when:
- The measurement of a subject is naturally discrete type, such as numbers resulted from dice rolled, count of people.
- The measurement is digitized machine data, which has no intermediate values between quantized levels due to sampling process.
- In later case, when resolution higher, the measurement is closer to analog/continuous signal/variable.
- Way of generate a PMF from discrete data:
- Plot a histogram of the data for all the $x$'s, the $y$-axis is the frequency or quantity at every $x$.
- Scale the $y$-axis by dividing with total number of data collected (data size) $longrightarrow$ and this is called PMF.
- Way of generate a PDF from discrete / continuous data:
- Find a continuous equation that models the collected data, let say normal distribution equation.
- Calculate the parameters required in the equation from the collected data. For example, parameters for normal distribution equation are mean and standard deviation. Calculate them from collected data.
- Based on the parameters, plot the equation with continuous $x$-value $longrightarrow$ that is called PDF.
- How to generate a CDF:
- In discrete case, CDF accumulates the $y$ values in PMF at each discrete $x$ and less than $x$. Repeat this for every $x$. The final plot is a monotonically increasing until $1$ in the last $x$ $longrightarrow$ this is called discrete CDF.
- In continuous case, integrate PDF over $x$; the result is a continuous CDF.
Why PMF, PDF and CDF?
- PMF is preferred when
- Probability at every $x$ value is interest of study. This makes sense when studying a discrete data - such as we interest to probability of getting certain number from a dice roll.
- PDF is preferred when
- We wish to model a collected data with a continuous function, by using few parameters such as mean to speculate the population distribution.
- CDF is preferred when
- Cumulative probability in a range is point of interest.
- Especially in the case of continuous data, CDF much makes sense than PDF - e.g., probability of students' height less than $170$ cm (CDF) is much informative than the probability at exact $170$ cm (PDF).
edited Mar 24 at 2:42
Rócherz
3,0263823
answered Mar 3 '14 at 10:14
user132704user132704
321133
$endgroup$
add a comment |
32
$begingroup$
Distribution Function
- The probability distribution function / probability function has ambiguous definition. They may be referred to:
- Probability density function (PDF)
- Cumulative distribution function (CDF)
- or probability mass function (PMF) (statement from Wikipedia)
- But what confirm is:
- Discrete case: Probability Mass Function (PMF)
- Continuous case: Probability Density Function (PDF)
- Both cases: Cumulative distribution function (CDF)
- Probability at certain $x$ value, $P(X = x)$ can be directly obtained in:
- PMF for discrete case
- PDF for continuous case
- Probability for values less than $x$, $P(X < x)$ or Probability for values within a range from $a$ to $b$, $P(a < X < b)$ can be directly obtained in:
- CDF for both discrete / continuous case
- Distribution function is referred to CDF or Cumulative Frequency Function (see this)
In terms of Acquisition and Plot Generation Method
- Collected data appear as discrete when:
- The measurement of a subject is naturally discrete type, such as numbers resulted from dice rolled, count of people.
- The measurement is digitized machine data, which has no intermediate values between quantized levels due to sampling process.
- In later case, when resolution higher, the measurement is closer to analog/continuous signal/variable.
- Way of generate a PMF from discrete data:
- Plot a histogram of the data for all the $x$'s, the $y$-axis is the frequency or quantity at every $x$.
- Scale the $y$-axis by dividing with total number of data collected (data size) $longrightarrow$ and this is called PMF.
- Way of generate a PDF from discrete / continuous data:
- Find a continuous equation that models the collected data, let say normal distribution equation.
- Calculate the parameters required in the equation from the collected data. For example, parameters for normal distribution equation are mean and standard deviation. Calculate them from collected data.
- Based on the parameters, plot the equation with continuous $x$-value $longrightarrow$ that is called PDF.
- How to generate a CDF:
- In discrete case, CDF accumulates the $y$ values in PMF at each discrete $x$ and less than $x$. Repeat this for every $x$. The final plot is a monotonically increasing until $1$ in the last $x$ $longrightarrow$ this is called discrete CDF.
- In continuous case, integrate PDF over $x$; the result is a continuous CDF.
Why PMF, PDF and CDF?
- PMF is preferred when
- Probability at every $x$ value is interest of study. This makes sense when studying a discrete data - such as we interest to probability of getting certain number from a dice roll.
- PDF is preferred when
- We wish to model a collected data with a continuous function, by using few parameters such as mean to speculate the population distribution.
- CDF is preferred when
- Cumulative probability in a range is point of interest.
- Especially in the case of continuous data, CDF much makes sense than PDF - e.g., probability of students' height less than $170$ cm (CDF) is much informative than the probability at exact $170$ cm (PDF).
edited Mar 24 at 2:42
Rócherz
3,0263823
answered Mar 3 '14 at 10:14
user132704user132704
321133
$endgroup$
add a comment |
32
32
32
$begingroup$
Distribution Function
- The probability distribution function / probability function has ambiguous definition. They may be referred to:
- Probability density function (PDF)
- Cumulative distribution function (CDF)
- or probability mass function (PMF) (statement from Wikipedia)
- But what confirm is:
- Discrete case: Probability Mass Function (PMF)
- Continuous case: Probability Density Function (PDF)
- Both cases: Cumulative distribution function (CDF)
- Probability at certain $x$ value, $P(X = x)$ can be directly obtained in:
- PMF for discrete case
- PDF for continuous case
- Probability for values less than $x$, $P(X < x)$ or Probability for values within a range from $a$ to $b$, $P(a < X < b)$ can be directly obtained in:
- CDF for both discrete / continuous case
- Distribution function is referred to CDF or Cumulative Frequency Function (see this)
In terms of Acquisition and Plot Generation Method
- Collected data appear as discrete when:
- The measurement of a subject is naturally discrete type, such as numbers resulted from dice rolled, count of people.
- The measurement is digitized machine data, which has no intermediate values between quantized levels due to sampling process.
- In later case, when resolution higher, the measurement is closer to analog/continuous signal/variable.
- Way of generate a PMF from discrete data:
- Plot a histogram of the data for all the $x$'s, the $y$-axis is the frequency or quantity at every $x$.
- Scale the $y$-axis by dividing with total number of data collected (data size) $longrightarrow$ and this is called PMF.
- Way of generate a PDF from discrete / continuous data:
- Find a continuous equation that models the collected data, let say normal distribution equation.
- Calculate the parameters required in the equation from the collected data. For example, parameters for normal distribution equation are mean and standard deviation. Calculate them from collected data.
- Based on the parameters, plot the equation with continuous $x$-value $longrightarrow$ that is called PDF.
- How to generate a CDF:
- In discrete case, CDF accumulates the $y$ values in PMF at each discrete $x$ and less than $x$. Repeat this for every $x$. The final plot is a monotonically increasing until $1$ in the last $x$ $longrightarrow$ this is called discrete CDF.
- In continuous case, integrate PDF over $x$; the result is a continuous CDF.
Why PMF, PDF and CDF?
- PMF is preferred when
- Probability at every $x$ value is interest of study. This makes sense when studying a discrete data - such as we interest to probability of getting certain number from a dice roll.
- PDF is preferred when
- We wish to model a collected data with a continuous function, by using few parameters such as mean to speculate the population distribution.
- CDF is preferred when
- Cumulative probability in a range is point of interest.
- Especially in the case of continuous data, CDF much makes sense than PDF - e.g., probability of students' height less than $170$ cm (CDF) is much informative than the probability at exact $170$ cm (PDF).
edited Mar 24 at 2:42
Rócherz
3,0263823
answered Mar 3 '14 at 10:14
user132704user132704
321133
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Distribution Function
- The probability distribution function / probability function has ambiguous definition. They may be referred to:
- Probability density function (PDF)
- Cumulative distribution function (CDF)
- or probability mass function (PMF) (statement from Wikipedia)
- But what confirm is:
- Discrete case: Probability Mass Function (PMF)
- Continuous case: Probability Density Function (PDF)
- Both cases: Cumulative distribution function (CDF)
- Probability at certain $x$ value, $P(X = x)$ can be directly obtained in:
- PMF for discrete case
- PDF for continuous case
- Probability for values less than $x$, $P(X < x)$ or Probability for values within a range from $a$ to $b$, $P(a < X < b)$ can be directly obtained in:
- CDF for both discrete / continuous case
- Distribution function is referred to CDF or Cumulative Frequency Function (see this)
In terms of Acquisition and Plot Generation Method
- Collected data appear as discrete when:
- The measurement of a subject is naturally discrete type, such as numbers resulted from dice rolled, count of people.
- The measurement is digitized machine data, which has no intermediate values between quantized levels due to sampling process.
- In later case, when resolution higher, the measurement is closer to analog/continuous signal/variable.
- Way of generate a PMF from discrete data:
- Plot a histogram of the data for all the $x$'s, the $y$-axis is the frequency or quantity at every $x$.
- Scale the $y$-axis by dividing with total number of data collected (data size) $longrightarrow$ and this is called PMF.
- Way of generate a PDF from discrete / continuous data:
- Find a continuous equation that models the collected data, let say normal distribution equation.
- Calculate the parameters required in the equation from the collected data. For example, parameters for normal distribution equation are mean and standard deviation. Calculate them from collected data.
- Based on the parameters, plot the equation with continuous $x$-value $longrightarrow$ that is called PDF.
- How to generate a CDF:
- In discrete case, CDF accumulates the $y$ values in PMF at each discrete $x$ and less than $x$. Repeat this for every $x$. The final plot is a monotonically increasing until $1$ in the last $x$ $longrightarrow$ this is called discrete CDF.
- In continuous case, integrate PDF over $x$; the result is a continuous CDF.
Why PMF, PDF and CDF?
- PMF is preferred when
- Probability at every $x$ value is interest of study. This makes sense when studying a discrete data - such as we interest to probability of getting certain number from a dice roll.
- PDF is preferred when
- We wish to model a collected data with a continuous function, by using few parameters such as mean to speculate the population distribution.
- CDF is preferred when
- Cumulative probability in a range is point of interest.
- Especially in the case of continuous data, CDF much makes sense than PDF - e.g., probability of students' height less than $170$ cm (CDF) is much informative than the probability at exact $170$ cm (PDF).
edited Mar 24 at 2:42
Rócherz
3,0263823
answered Mar 3 '14 at 10:14
user132704user132704
321133
edited Mar 24 at 2:42
Rócherz
3,0263823
edited Mar 24 at 2:42
Rócherz
3,0263823
edited Mar 24 at 2:42
Rócherz
3,0263823
3,0263823
answered Mar 3 '14 at 10:14
user132704user132704
321133
answered Mar 3 '14 at 10:14
user132704user132704
321133
answered Mar 3 '14 at 10:14
user132704user132704
321133
321133
add a comment |
add a comment |
0
$begingroup$
Some abuse of language exists in these terms, which can vary. Below is a common usage.
In the continuous case (density):
(continuous) probability distribution function = probability density function = density function
(continuous) probability distribution = density
In the discrete case (mass/distribution):
(discrete) probability distribution function = probability mass function
(discrete) probability distribution = distribution
Oddly enough, you may never see a probability mass function called a mass function or a distribution function, nor may you see a discrete probability distribution called a mass. I am sure there is some historical reason as to why. As they say, das war schon immer so und wird auch immer so bleiben.
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
$endgroup$
add a comment |
0
$begingroup$
Some abuse of language exists in these terms, which can vary. Below is a common usage.
In the continuous case (density):
(continuous) probability distribution function = probability density function = density function
(continuous) probability distribution = density
In the discrete case (mass/distribution):
(discrete) probability distribution function = probability mass function
(discrete) probability distribution = distribution
Oddly enough, you may never see a probability mass function called a mass function or a distribution function, nor may you see a discrete probability distribution called a mass. I am sure there is some historical reason as to why. As they say, das war schon immer so und wird auch immer so bleiben.
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
$endgroup$
add a comment |
0
0
0
$begingroup$
Some abuse of language exists in these terms, which can vary. Below is a common usage.
In the continuous case (density):
(continuous) probability distribution function = probability density function = density function
(continuous) probability distribution = density
In the discrete case (mass/distribution):
(discrete) probability distribution function = probability mass function
(discrete) probability distribution = distribution
Oddly enough, you may never see a probability mass function called a mass function or a distribution function, nor may you see a discrete probability distribution called a mass. I am sure there is some historical reason as to why. As they say, das war schon immer so und wird auch immer so bleiben.
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
$endgroup$
Some abuse of language exists in these terms, which can vary. Below is a common usage.
In the continuous case (density):
(continuous) probability distribution function = probability density function = density function
(continuous) probability distribution = density
In the discrete case (mass/distribution):
(discrete) probability distribution function = probability mass function
(discrete) probability distribution = distribution
Oddly enough, you may never see a probability mass function called a mass function or a distribution function, nor may you see a discrete probability distribution called a mass. I am sure there is some historical reason as to why. As they say, das war schon immer so und wird auch immer so bleiben.
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
answered Feb 21 at 4:38
Adam EricksonAdam Erickson
1114
1114
add a comment |
add a comment |
protected by Zev Chonoles Aug 23 '16 at 9:24
Thank you for your interest in this question.
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2
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The density (when it exists) is the derivative of the distribution function.
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– Joel Cohen
Jul 27 '12 at 13:31
1
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You mean, "Difference between Probability density function and cumulative distribution function?"?
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– Matt O'Brien
Feb 5 '14 at 21:08