Types of Probability Density Function

Types of probability density function in probability refer to different types of functions that are used to describe a continuous probability distribution. A probability mass function, on the other hand, is used to describe a discrete probability distribution and is analogous to the probability density function.

As there are different types of probability distribution functions thus, there are corresponding types of probability density functions. In this article, we will learn more about different types of probability density function, their use, formulas and see associated examples.

The types of probability density function are used to describe distributions like continuous uniform distribution, normal distribution, Student t distribution, etc. The probability density function gives the probability that the value of a random variable will fall between a range of values. A cumulative distribution function and the probability density function are used to describe a continuous distribution.

Probability Density Function Definition

A probability density function can be defined as a function that gives the likelihood of occurrence of a random variable between a given interval of values. To determine this probability, the probability density function has to be integrated between the two specified limits. Suppose the probability that a random variable, X, lies between points a and b has to be determined then the general formula is given as follows:

P(a < X ≤ b) = F(b) - F(a) = \(\int_{a}^{b}f(x)dx\)

There are two important properties followed by a probability density function, f(x). These are given as follows:

• f(x) ≥ 0. The probability density function for all real numbers will always be positive.
• \(\int_{-\infty }^{\infty}f(x)dx\) = 1. The total area under the probability density curve has to be equal to 1.

There are several types of continuous distributions thus, different probability density functions are used. Some important continuous distributions are given below:

• Normal Distribution
• Standard Normal Distribution
• Student - t Distribution
• Chi-Square Distribution
• Continuous Uniform Distribution

The next section covers the probability density function formula for these distributions.

Normal Distribution Probability Density Function

A random variable that follows a normal distribution is denoted as \(X \sim N(\mu ,\sigma ^{2})\). Here, \(\mu\) is the mean and \(\sigma\)² is the variance and they form the parameters of the normal distribution. The graph of a normal distribution is a bell curve and is symmetric about the mean. The formula for the type of probability density function for a normal distribution is given below:

f(x) = \(\frac{1}{\sigma \sqrt{2\Pi}}e^{\frac{-1}{2}\left ( \frac{x - \mu }{\sigma } \right )^{2}}\)

Standard Normal Distribution Probability Density Function

When a continuous random variable, X, follows a normal distribution such that the mean equals 0 and the standard deviation is equal to 1 then such a probability distribution is known as a standard normal distribution. It is denoted as \(X \sim N(0 ,1)\)The formula for the probability density function of a standard normal distribution is given as follows:

f(x) = \(\frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}\)

Student T Distribution Probability Density Function

A continuous random variable following a student t distribution is denoted as \(X \sim t(\nu)\), where \(\nu\) denotes the degrees of freedom. A student t distribution is used when the sample size is very small and a normal distribution cannot be used. The probability density function of a student t distribution is given as follows:

f(x) = \(\frac{\Gamma \left ( \frac{\nu+1}{2} \right )}{\sqrt{\nu\pi}\Gamma \left ( \frac{\nu}{2} \right )}\left ( 1+\frac{x^{2}}{\nu} \right )^{-\frac{\nu+1}{2}}\)

Here, \(\Gamma\) represents the gamma function

Chi-Squared Distribution Probability Density Function

Chi squared distribution is widely used for hypothesis testing. It can be defined as the sum of squares of k independent standard normal variables. It is denoted as \(X \sim \chi^{2}(k)\). The type of probability density function used for a chi squared distribution is given as follows:

f(x) = \(\frac{x^{\frac{k}{2}-1}e^{-\frac{x}{2}}}{2^{\frac{k}{2}}\Gamma\left ( \frac{k}{2} \right )}\), x > 0.

f(x) = 0, otherwise

Continuous Uniform Distribution Probability Density Function

A uniform distribution is used to describe a random experiment such that the outcome lies between two values. The notation is given as \(X \sim U(a, b)\). The probability density function when x lies between a and b is given as follows:

f(x) = 1 / (b - a)

Related Articles:

• Probability Theory
• Probability Rules
• Probability and Statistics

Important Notes on Types of Probability Density Function

• Types of probability density function help to describe the probability distributions followed by different continuous random variables.
• There are 5 main types of probability density functions depending upon the different distributions such as normal distribution, standard normal distribution, Student - t distribution, chi-square distribution, and continuous uniform distribution.
• A probability density function will always be positive and the total area under the curve will always be equal to 1.

What are the Types of Probability Density Function?

There are mainly 6 types of probability density function in probability theory. These are used for the following distributions:

• Normal Distribution
• Standard Normal Distribution
• Student - t Distribution
• Chi-Square Distribution
• Continuous Uniform Distribution

What Type of Probability Density Function is Used for a Normal Distribution?

A normal distribution is represented as \(X \sim N(\mu ,\sigma ^{2})\). The formula for the type of probability density function used for a normal distribution is given by f(x) = \(\frac{1}{\sigma \sqrt{2\Pi}}e^{\frac{-1}{2}\left ( \frac{x - \mu }{\sigma } \right )^{2}}\)

What Type of Probability Density Function is Used for a Standard Normal Distribution?

A standard normal distribution is a type of normal distribution where \(\mu\) = 0 and \(\sigma\)² = 1. The formula for this type of probability density function is given as f(x) = \(\frac{1}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}\).

What is the Difference Between a Probability Density Function and a Probability Mass Function?

A probability density function is used for continuous random variables and gives the probability that the value of this variable will lie between a particular interval. The probability mass function gives the probability that a discrete random variable will be exactly equal to some number.

What is the Most Common Type of Probability Density Function?

The most commonly used type of probability density function is the normal distribution because most continuous data follows a bell-shaped curve when compiled and graphed.

Can a Binomial Distribution Have a Probability Density Function?

A binomial distribution is a type of discrete probability distribution. Thus, a probability mass function will be used to describe its characteristics rather than a probability density function.

What are the Two Important Properties that Apply to All Types of Probability Density Functions?

All types of probability density functions follow two important properties. These are listed below:

• The total area under the probability density function curve will always be equal to 1.
• The probability density function cannot be negative.