Whats the difference between probability density function and probability distribution function?
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2$\begingroup$ The density (when it exists) is the derivative of the distribution function. $\endgroup$– Joel CohenJul 27, 2012 at 13:31
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2$\begingroup$ You mean, "Difference between Probability density function and cumulative distribution function?"? $\endgroup$– tumultous_roosterFeb 5, 2014 at 21:08
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2$\begingroup$ Does this answer your question? Concept of Probability distribution $\endgroup$– user53259Jan 1, 2022 at 6:53
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3 Answers
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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).
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2$\begingroup$ Note that the probability at P(X=x) cannot be obtained in the continuous case by evaluating the density function at x. Rather you can obtain it by integrating the density function from x to x, which will return 0. This is because the probability of observing any particular value in a continuous distribution is 0. $\endgroup$– SamJul 25, 2021 at 22:10
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The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is...
if $f$ is discrete: $$ F(k) = \sum_{i \le k} f(i) $$
if $f$ is continuous: $$ F(x) = \int_{y \le x} f(y)\,dy $$
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$\begingroup$ what is meant by discrete and continuous? $\endgroup$ Jul 27, 2012 at 13:40
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$\begingroup$ @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. $\endgroup$ Jul 27, 2012 at 13:45
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$\begingroup$ @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. $\endgroup$ Jul 27, 2012 at 13:47
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1$\begingroup$ 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! $\endgroup$ Jul 27, 2012 at 13:50
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$\begingroup$ @maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition. $\endgroup$ Jul 27, 2012 at 14:12
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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.
