Probability density function

Probability density function (PDF) is a statistical expression that defines a probability distribution for a continuous random variable as opposed to a discrete random variable. When the PDF is graphically portrayed, the area under the curve will indicate the interval in which the variable will fall. The total area in this interval of the graph equals the probability that a random variable will fall in that range.

Definition

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

Properties

The probability density function of a continuous random variable must satisfy two properties:

1. The function must be non-negative everywhere, i.e., for all x in the real numbers, f(x) ≥ 0.
2. The total area under the curve must equal to 1.

Examples

Some examples of probability density functions are:

• The exponential distribution
• The normal distribution
• The chi-squared distribution
• The Student's t-distribution

See also

• Cumulative distribution function
• Probability mass function
• Probability distribution
• Random variable

References

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