# Difference between Probability density function and distribution function?

i am learning for my statistics exam and have to know a lot of theory. My question is:

Whats the difference between Probability density function and distribution function?

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The density (when it exists) is the derivative of the distribution function. –  Joel Cohen Jul 27 '12 at 13:31
Thx for your answer! Yes thats clear for me!!! But, where do you use it and especially why? –  Le Chifre Jul 27 '12 at 13:36
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 $$F(k) = \sum_{i \le k} f(i)$$ if $f$ is discrete and $$F(x) = \int_{y \le x} f(y)\,dy$$ if $f$ is continuous.

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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) --> 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 --> that is called PDF
4. Way of 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 --> this is called discrete CDF
• In continuous case, integrate PDF over x, results 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)
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