What is the probability density function (PDF)?

The probability density function describes the likelihood of each specific value that a variable can have.

Example of a discrete PDF

For a discrete variable, the PDF is a list containing each value that the variable can have and its associated probability. For example, a candy manufacturer produces a single type of candy in multiple colors. 30% of the candies produced are yellow, 10% are orange, 10% are red, 20% are green, and 30% are blue.
Discrete PDF

This bar chart displays the PDF for candy color. Each bar represents the probability of candies of that color expressed as a percentage.

Example of a continuous PDF

For a continuous variable, the PDF is the curve that approximates the shape when its values are displayed on a bar chart or histogram. For example, a machine that cuts corks for wine bottles produces corks with different diameters. In the following bar chart of cork diameters, each bar represents the percent of corks with that corresponding diameter.
Continuous PDF

The curve is the PDF for cork diameter. Use the PDF to determine the probability of an event occurring. For example, only a small percentage of corks (1%) have a diameter less than 2.8 cm.

Continuous PDF with specification limits

If the specification limits for cork diameter are 2.85 cm to 3.15 cm, the PDF gives the probability or percentage of all corks from this process that meet the specifications.

The shape of the PDF is different for different distributions. The familiar bell-shaped curve represents the PDF for a normal distribution. While cork diameter follows a normal distribution, other measurements, such as the force it takes to pull the cork out of the wine bottle, may follow a different distribution. For example, the PDF for a log-normal distribution has a long right tail.
Lognormal PDF

Because a wine bottle occasionally requires an unusual amount of force to remove the cork, measures of this force often follow a distribution with a long right tail such as the lognormal distribution.

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