What is a probability density function (p.d.f.)?

Introduction to Probability Density Function (p.d.f.)

A probability density function (p.d.f.) relates to continuous random variables, which can take on any value within a given range.
The p.d.f. is the counterpart in the continuous world to the probability mass function in the discrete world.
The p.d.f. describes the likelihood of each possible outcome for a continuous random variable.

Properties of Probability Density Function

The function must be non-negative, meaning its value cannot be less than zero for all possible outcomes.
The total area under the curve of the function must equal 1, representing the total probability of all possible outcomes.

Using Probability Density Function

To find the probability of a continuous random variable taking a value in a certain interval, one needs to calculate the area under the function curve over that interval.
Although you cannot get the probability for a specific point from a p.d.f. (since the area under the point is zero), you can calculate the probability over a range of values.
The p.d.f. allows you to determine the likelihood of the random variable falling within a certain range.

Expectation and Variance

Expectation or mean of a random variable can be calculated by integrating the product of the random variable and the p.d.f.
Variance or spread of a random variable can also be calculated using the p.d.f., by integrating the square of the random variable times the p.d.f., and then subtracting the square of the mean value.

Interpreting Probability Density Functions

High peaks in the p.d.f. indicate values that are more likely to occur.
A flatter p.d.f indicates that all values are approximately equally likely.
The shape of the p.d.f. gives a complete picture of the probability distribution of a continuous random variable.
The width of the p.d.f. provides information about the uncertainty of the random variable.
Understanding the properties of the p.d.f. is key to interpreting the behaviour of continuous random variables.

Note: Always remember that although interpreting a p.d.f. graphically is useful, the actual calculation of probabilities requires integrating the function.