Introduction to Probability Density Function [Formula, Properties, Applications, Examples]

Probability Density Function (PDF) is an expression in statistics that denotes the probability distribution of a discrete random variable. Probability distribution, in simple terms, can be defined as a likelihood of an outcome of a random variable like a stock or an ETF. Discrete variables occur in contrast to a continuous random variable whose accurate value can be determined.

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For instance, the value of scrip in a stock market has only two decimal points (for example, 65.76) in a discrete random variable instead of a continuous variable with any number of decimal points (example: 65.7685434567). 

A probability density function is a statistical tool used to determine the likelihood of the outcome of a discrete random variable. When plotted on a graph, PDFs look identical to a bell curve in which the area under the curve represents the probability of the outcome. 

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When projected as a graphic model, the area under the curve represents the range in which the values of the discrete random variables will fall. Thus, the total area under the curve is equal to the probability of the variable’s outcome. 

The probability density function can determine the likelihood of a random variable falling within a specific range of values.

Typically, probability density functions analyse the risks and potential revenue associated with a specific fund in the stock market.

Conditions to be Satisfied by a function to be considered a Probability Density Function

The value of a discrete variable can be accurately measured in contrast to a continuous variable that can have an infinite number of values. Any function should satisfy the below two conditions to be a probability density function: 

  • The f(x) value for each possible value of the random variable should be positive (non-negative).
  • The integral value of the total area of the curve (integral of all possible values of the random variable) should be 1.

Difference between Probability Density Function and Probability Distribution Function

Random variables can have many values. The description of each possible value that a random variable can have is called its probability distribution. 

The probability distribution gives a set of outcomes and their related probabilities. The statistical function that represents a continuous probability distribution is known as the probability density function. 

There is another statistical tool that represents a discrete probability distribution called the probability mass function. This gives a detailed account of all the possible outcomes and their likelihood probabilities. 

Expression for Probability Density Functions

If the random variable is discrete, its probability distribution is called probability mass function, and if it is a continuous variable, the probability distribution is called probability density function.

A PDF is used when the random variable in question has a range of possible values. Their probability distribution is used to determine the exact value.

Let the random variable be denoted by X. The probability density function, f of the random variable X can be expressed as     

  • The value of the random variable lies between a and b.
  • If X denotes the probability of selecting a particular number from the range (interval) r and s, then the probability density function can be expressed as

             f(x) = 1/(s − r) for r < x < s and f(x) = 0 for x < r or x > s

  • The PDF F is represented as:

F(x) = P{X ≤ x

which is called the distribution function or the cumulative distribution function of X.

Considering the random variable X has a probability distribution function f(x), then the relationship between f and F can be established as 

F′(.x) = f(x)

The distribution function of a discrete random variable is different from its probability distribution function. The relationship between the two can be expressed as below:

The expectation of the random variable is denoted as,

Thus, all discrete and random variables can be treated uniformly with the help of a combined theory.

The formula of Probability Density Function

The probability of a continuous random variable X on some fixed value x is always 0. In this case, P(X = x) cannot be used. The value of the X lying between a range of values (a,b) should be determined. To determine the same, the following formula is used.

Properties of a Probability Density Function 

A continuous random variable that takes its value between the range (a,b), for instance, will be estimated by calculating the area under the curve and the X-axis plotted with (a) as its lower limit and (b) as its upper limit. The probability density function for the above is represented as:

The probability density function is positive (non-negative) for all possible values. This means f(x)≥ 0, for every x. The area falling between the density curve and the X-axis (horizontal axis) equals 1. 

This can also be denoted as:


The density function curve is continuous throughout the given range, which is clearly defined against a series of continuous values or the variable’s domain. 

Applications of Probability Density Function

  • The probability density function is used in the yearly modelling of atmospheric NO concentration levels.
  • Modelling of diesel engine combustion.
  • In statistics, the probability density function is used to determine the possibilities of the outcome of a random variable. 

Examples of Probability Density Function

  • Example 1

Below is an example of how probability density function (PDF) is used to determine the risk potential of an investor in the stock market:

First, PDFs are generated as a graphic tool based on historical information. 

The most common form of PDF is the neutral projection, where the risk is equal to the reward across a range of possibilities. Investors with less risk-taking capability will only be rewarded with limited profits, and hence they come under the left side of the bell curve. Conversely, investors with high risk-taking abilities are likely to be rewarded with higher yields, and therefore, fall under the right side of the curve.

Most of the investors fall under average risk-taking ability, and hence they occupy the middle of the curve.

This helps in analysing the category of investors based on the data received. This helps stock market brokers to identify their target category of customers to sell their products. 

  • Example 2

One of the essential applications of the probability density function is the Gaussian random variable, also known as a normal random variable.

In both cases, the graph gives a bell curve for the probability density function.

The density can be expressed as 

The graph of the above density equation is given below.

The area under the curve represents the actual value of the Gaussian random variable.

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Can a Probability Density Function be greater than 1?

As the probability function gives a fixed probability, it cannot be more than 1. A PDF f(x), however, can have values greater than 1 for certain values of X. This can happen as they represent the probable values (range for the area under the curve) and not the exact values of f(x).

What can be inferred from the probability density function?

The probability density function is the statistical technique used to determine the possibility of the outcome of a discrete random variable. The PDFs are depicted on a graph with the background data plotted in X and Y axes. The graph gives out a bell curve. The range of the curve gives us the range of the possible values, and the area under the curve provides the exact value of the discrete random variable.

What will be the probability density function of normal distribution?

A normal distribution is symmetric and has a non-zero probability for all positive and negative values of the random variable. The non-zero probability holds good even if the probability is assigned to values with more than 3 or 4 standard deviations as the mean is negligible.

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