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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)