Probability Density Functions: A Beginner’s Guide to Understanding Probability Distributions

Source: https://www.simplilearn.com/tutorials/statistics-tutorial/probability-density-function

In statistics and probability theory, a probability density function (PDF) is a function that describes the probability of a continuous random variable taking on a particular value. In this article, we will take a closer look at probability density functions and how they are used to represent probability distributions in continuous data.

What is a probability density function?

A probability density function is a mathematical function that describes the probability of a continuous random variable taking on a particular value. It is defined as a function that satisfies the following conditions:

1. The probability of a continuous random variable falling within a particular range of values is equal to the integral of the probability density function over that range.
2. The probability density function is non-negative for all values of the random variable.
3. The total probability of the random variable taking on any value is equal to 1.

In other words, a probability density function describes the relative likelihood of a continuous random variable taking on different values. It is used to represent the probability distribution of a continuous random variable and can be used to calculate probabilities of different events or ranges of values.

Example:

To illustrate how a probability density function is used to represent a probability distribution, let’s consider an example. Suppose we are interested in studying the height of adult males in a population. We can represent the probability distribution of height using a probability density function. Let’s assume that the heights of adult males in this population are normally distributed, with a mean of 68 inches and a standard deviation of 3 inches. The probability density function for this distribution can be represented by the following equation:

f(x) = 1/(3sqrt(2pi)) * exp(-1/2*((x-68)/3)²)

This equation describes the probability of an adult male in this population having a height of x inches. For example, if we wanted to calculate the probability of an adult male in this population having a height between 66 and 68 inches, we would take the integral of the probability density function over this range:

P(66 <= x <= 68) = integral of f(x) from 66 to 68

Using this equation, we can calculate the probability of different events or ranges of values occurring in the population.

Conclusion:

Probability density functions are mathematical functions that describe the probability of a continuous random variable taking on a particular value. They are used to represent the probability distribution of continuous data and can be used to calculate probabilities of different events or ranges of values. By understanding probability density functions and how they are used to represent probability distributions, we can gain valuable insights into the patterns and relationships in our data.