What is the difference between "probability density function" and "probability distribution function"? Whats the difference between probability density function and probability distribution function? 
 A: 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).


A: 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:
$$
F(x) = \int_{y \le x} f(y)\,dy
$$
A: 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.
