# What is the difference between “probability density function” and “probability distribution function”?

Whats the difference between probability density function and probability distribution function?

• The density (when it exists) is the derivative of the distribution function. – Joel Cohen Jul 27 '12 at 13:31
• You mean, "Difference between Probability density function and cumulative distribution function?"? – Matt O'Brien Feb 5 '14 at 21:08

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

• what is meant by discrete and continuous? – Le Chifre Jul 27 '12 at 13:40
• @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. – Robert Mastragostino Jul 27 '12 at 13:45
• @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. – August Karlstrom Jul 27 '12 at 13:47
• 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! – Le Chifre Jul 27 '12 at 13:50
• @maximus That's correct though you may have to count forever. Check out the concept of a countable set for an exact definition. – August Karlstrom Jul 27 '12 at 14:12

Distribution Function

1. 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)
2. But what confirm is:
• Discrete case: Probability Mass Function (PMF)
• Continuous case: Probability Density Function (PDF)
• Both cases: Cumulative distribution function (CDF)
3. Probability at certain $$x$$ value, $$P(X = x)$$ can be directly obtained in:
• PMF for discrete case
• PDF for continuous case
4. 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
5. Distribution function is referred to CDF or Cumulative Frequency Function (see this)

In terms of Acquisition and Plot Generation Method

1. 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.
2. 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.
3. 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.
4. 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?

1. 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.
2. 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.
3. 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).

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.

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