Probability density function

Boxplot and probability density function of a normal distribution N(0, σ²).

A probability density function is a function that can be defined for any continuous probability distribution. The integral of the probability density function in the interval $[a,b]$ yields the probability that a given random variable with the given density is contained in the interval provided.

The probability density function is necessary to be able to work with continuous distributions. Casting a dice will give the numbers 1 to 6, with a probability of ${\tfrac {1}{6}}$, but this is not a continuous function, as only the numbers 1 to 6 are possible. In contrast, two people will not have the same height, or the same weight. Using a probability density function, it is possible to determine the probability for people between 180 centimetres (71 in) and 181 centimetres (71 in), or between 80 kilograms (176.4 lb) and 81 kilograms (178.6 lb), even thought there are infinitely many values between these two bounds.