Probability Density Function : Meaning, Formula, and Graph
Last Updated :
23 Jul, 2025
What is the Probability Density Function?
Probability Density Function (PDF) and Cumulative Distribution Function (CDF) describe the probability distribution of a continuous random variable. In simpler terms, PDF tells about how likely different values of the continuous random variable are. By differentiating the CDF of a continuous random variable, we can determine the Probability Density Functions. In the same way, if we integrate the Probability Density Function, we can obtain the probability or CDF between two specific points.
For example, Suppose there is a continuous random variable with PDF given by f(x) = x + 4, where 0 < x ≤ 4; and we want to determine P(1.5 < X < 2). To find the probability of the given continuous random variable, we will integrate x + 4 within the limits of 1.5 and 2. The result will be 2.875. Hence, the probability of X ranging between 1.5 and 2 is 2.875.
Probability Density Function Formula
Function Formula in case of Discrete Random Variables
- The PDF of discrete random variables gives the probability of the variable taking a specific value.
- The PDF formula is denoted as P(X=x), where X is the random variable, and x is a particular value it can take.
- A function can be quantified as a Probability Function or Probability Density Function of a discrete random variable, when
fX(x) ≥ 0, for all x within the range of X
\sum_xf_X(x)=1
Example:
Suppose there are 6 balls in a bag. The random variable X is the weight of a ball (in kg) selected at random. Balls 1, 2, and 3 weighs 0.5 kg; Balls 4 and 5 weighs 0.25 kg; and Ball 6 weighs 0.3 kg. Write the Probability Density Function for X.
Solution:
fX(0.5) = P(X = 0.5) = \frac{3}{6}=\frac{1}{2}
fX(0.25) = P(X = 0.25) = \frac{2}{6}=\frac{1}{3}
fX(0.3) = P(X = 0.3) = \frac{1}{6}
Function Formula in case of Continuous Random Variables
- The PDF of continuous random variables gives the likelihood of the variable falling within a specific interval.
- The PDF is denoted as f(x), where x is the continuous random variable.
- The probability of x lying within a given interval [a,b] is calculated by integrating the PDF over that range. Therefore,
P(a<x<b)=\int_{a}^{b}f_X(x)~dx
- A function will serve as a PDF when it follows the following conditions:
fX(x) ≥ 0, -∞ ≤ x ≤ ∞
\int_{-\infty}^{\infty}f_X(x)~dx=1
Here the limits are taken as -∞, ∞. However, in real life, we will use the values of x, applicable to the question. It means that if the variable is given between the range of a and b (a < x < b), then the limits for integration will be a and b.
Example:
A continuous random variable Y has PDF fY(y) = 10y2(1-y), where 0 < y < 2. Determine P(Y < 0.2).
Solution:
P(Y < 0.2) = \int_{0}^{0.2}10y^2(1-y)dy
P(Y < 0.2) = [\frac{5y^2(4-3y)}{6}]^{0.2}_{0}
P(Y < 0.2) = 0.023
Probability Density Function Graph
The probability density function (PDF) is found by adding up the density of the variable over a certain range. We use the symbol f(x) to represent this function. At any point on the graph, the value of this function is positive or zero. When we calculate the definite integral of the PDF over the entire range, the result is always one.
The graph of PDFs often looks like a bell curve, with the likelihood of outcomes shown below the curve. The figure below illustrates the graph of a probability density function for a continuous random variable x, with the function represented by f(x).

Properties of Probability Density Function
1. The density function, denoted as f(x), is used for a continuous random variable with values between specific limits, a and b. To find the Probability Density Function (PDF), we calculate the area under the curve between these limits on the X-axis.
2. The PDF, f(x), is always greater than or equal to zero for any possible value of x.
3. The total area under the density curve and the X-axis, within the given range from a to b, is equal to 1. This represents the entire probability space for the continuous random variable.
4. The density function curve extends smoothly over the entire specified range, illustrating the continuous nature of the random variable.
5. The PDF is defined across a range of continuous values, reflecting the variable's domain.
Probability Distribution Function of Discrete Distribution
I. Discrete Uniform Distribution
The Discrete Uniform Distribution represents outcomes where each possible value has an equal chance of occurring. For example, rolling a fair six-sided die is a classic case of discrete uniform distribution.
P(X = x) = \frac{1}{n}
The Bernoulli Distribution is applicable to situations with two possible outcomes, typically labeled as success and failure. It is often used in scenarios like coin flips, where success might be getting heads.
P(X = x) = px (1-p)1-x, x = 0, 1; 0 < p < 1
Binomial Distribution deals with the number of successes in a fixed number of independent Bernoulli trials. For instance, determining the probability of getting a certain number of heads in multiple coin flips.
P(X = x) = nCx px (q)n-x, x = 0, 1, 2,......,n; 0 < p < 1
IV. Geometric Distribution
Geometric Distribution models the number of trials needed for the first success in a sequence of independent Bernoulli trials. For example, finding the probability of the first successful free throw in basketball.
P(X = x) = (1 - p)k-1.p
The Negative Binomial Distribution focuses on the number of trials needed for a fixed number of successes in a sequence of independent Bernoulli trials. It's applicable to scenarios like predicting the number of attempts to make three successful shots in basketball.
P(X=x)=^{x-1}C_{k-1}\theta^k(1-\theta)^{x-k}~~here,x=k,k+1,...,0<\theta<1
The Poisson Distribution is useful for events with a known average rate of occurrence within a fixed interval. It's commonly employed in areas, such as predicting the number of emails received in an hour.
P(X=x)=\frac{\lambda{^x}e^{-\lambda}}{x!},x=1,2,3,........;\lambda{>0}
Probability Distribution Function of Continuous Distribution
I. Continuous Uniform Distribution
The Continuous Uniform Distribution represents a constant probability for all values within a specified range. The Probability Distribution Function (PDF) for this distribution is a flat, horizontal line, indicating equal likelihood for any value within the given interval.
f(x)=\frac{1}{b-a},~for~a\leq{x}\leq{b}
where a and b are the lower and upper limits of the distribution.
The Gamma Distribution is used to model the time until a certain number of events occur in a Poisson process. Its PDF is characterized by a shape parameter and a rate parameter, providing flexibility in describing various scenarios, including waiting times and reliability analysis.
f_X(x)=\frac{\lambda^\alpha}{Γ(\alpha)}x^{\alpha-1}e^{-\lambda x},for~x>0
The Exponential Distribution is often employed to model the time between events in a Poisson process. Its PDF exhibits a decreasing exponential curve, emphasizing a constant hazard rate and frequent use in reliability and queuing studies.
f(x;\lambda)=\lambda{e}^{-\lambda x},for~x\geq{0}
The Chi-Square Distribution is widely utilized in statistical hypothesis testing. Its PDF is determined by the degrees of freedom, influencing the shape of the distribution. Common applications include testing variance and goodness of fit.
f_x(x)=\frac{(\frac{1}{2})^{\frac{1}{2}\nu}}{Γ(\frac{1}{2}\nu)}x^{\frac{1}{2}\nu-1}e^{\frac{-1}{2}x},x>0
V. Beta Distribution
The Beta Distribution is versatile, representing random variables bounded within a specific interval. Its PDF is defined by shape parameters, allowing it to model a range of outcomes, including proportions and probabilities.
f_X(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{(\beta-1)}~for~0<x<1
The Lognormal Distribution is suitable for variables that follow a log-normal pattern. Its PDF is characterized by parameters influencing the shape and scale of the distribution. Applications include modelling stock prices and certain biological processes.
f(x)=\frac{1}{xσ√2π}e^\frac{-1}{2}(\frac{logx-μ}{σ})^2, for ~0<x<\infty
The Normal Distribution, or Gaussian Distribution, is a fundamental distribution found in many natural phenomena. Its PDF is the famous bell-shaped curve, determined by mean and standard deviation parameters. It is extensively used in statistical analyses due to the Central Limit Theorem and its prevalence in various fields.
f(x)=\frac{1}{σ\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-μ}{σ}\right)^2} , -∞ < x < ∞
Applications of Probability Density Function
Probability Density Function can be used in various fields and problems:
- Probability Assessment: It helps in assessing the probability of a continuous random variable falling within a specific range of values.
- Statistical Analysis: The PDF is a crucial tool in statistical analysis, providing insights into the distribution of continuous data.
- Area under the Curve: The area under the PDF curve between two points represents the probability of the variable lying within that range.
- Comparisons: It enables comparisons between different continuous random variables that help in understanding their respective likelihoods.
- Research and Sciences: PDF is widely used in fields like physics, engineering, and finance for modelling and analyzing continuous phenomena.
- Integration in Calculations: The PDF is often used in calculus, where integrating it over a range yields the probability of the random variable falling within that interval.
- Probability Density Assessment: It allows for the assessment of how densely the probability is distributed across different values of a continuous random variable.
- Understanding Distribution Shapes: The shape of the PDF curve provides insights into the characteristics of the underlying probability distribution.
- Risk Analysis: In finance and insurance, PDF is used to assess the probability of different financial outcomes and potential risks.
- Mathematical Basis: It serves as a fundamental concept in probability theory, providing a mathematical foundation for understanding continuous random variables.
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operationsâAddition, Subtraction, Multiplication, and Divisionâused for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 â it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (â)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice