What is Probability Density Function (PDF)?
Definition of Probability Density Function-
The probability function which is represented for the density of a continuous random variable lying between a certain range of values is known as Probability Density Function.
It is known as a probability distribution function or just a probability function. This function is referred to as a cumulative distribution function or Probability Mass Function (PMF).
Probability Distribution function (PDF) is defined for continuous random variables whereas PMF (Probability Mass Function) is defined for discrete random variables.
PMF does not work for continuous random variables because for any continuous random variable P(X=x) =0 for all x∈R , where R = real numbers.
The function defined in the form of an integral of the density of the variable density over a given range is known as PDF and it is generally denoted by f (x).
The function f(x) can be positive or non-negative at any point in the graph.
The integral of the Probability Density Function over the entire space is always equal to one.
Formula for Probability Density Function (PDF)-
The probability taken by a variable X in case of a continuous random variable on some given value x is always equal to zero. Instead of finding P(X = x), we need to calculate the probability of X lying in a given interval (a, b).
Now, let’s calculate it for P (a< X< b). We can do this by using the Probability Density Function. The Probability distribution function formula can be defined as,
P (a<X<b)= \[\int_{a}^{b}\]f(x)
The symbol f(x) is used to represent the curve. f(x) is the function that corresponds to the graph; to draw the graph of the probability distribution we use the density function f(x).
Formula Review-
Properties of Probability Density Function-
Let us assume x is the continuous random variable with f(x) as the density function, the probability distribution function should satisfy the following given conditions:
For a continuous random variable that takes any value between certain limits, let us say a and b, and we calculate it by finding the area under its curve and the x-axis, within(b) as the upper limit and the (a) as the lower limit and, then the Probability Density Function can be denoted by
P(x) = \[\int_{a}^{b}\]f(x)
It is non-negative for all the possible values, that is f(x) is greater than 0, for all values of x.
f(x) > 0
The area between the horizontal X-axis and the density curve is equal to 1, that is,
\[\int_{-∞}^{∞}\]dx = 1
Because of the property of continuous random variable, for all over the given range the density function curve is continuous which defines itself over a range of continuous values or the domain of the variable.
Types of Probability Distribution-
Probability Distributions generally come in many different shapes with various different properties, as defined by the standard deviation, mean, skewness, and kurtosis. The different Probability distributions represent different data generation processes and serve different purposes. There are different types of Probability Distribution like -
Normal Distribution
Chi square Distribution
Binomial Distribution
Poisson Distribution
Out of all the Probability Distributions, the normal distribution is the most commonly used in engineering, science, finance, and investing.
A Few Applications of Probability Density Function You Need to Know!
The applications of the probability density function are listed below:
The probability density function can be used to model the annual data of atmospheric NOx temporal concentration.
Probability Density Function is also used to model the diesel engine combustion.
In Statistics, it is used to calculate the probabilities associated with the random variables.
Probability Distributions are used in risk management.
It is also used to evaluate the probability and amount of losses.
Questions to be Solved-
Question 1) Let X be a random variable with Probability Density Function,
f(x) =
Find the constant c.
Solution) Now to find the constant c we can use \[\int_{-∞}^{∞}\]f(x)u du = 1
= \[\int_{-∞}^{∞}\]f(x)u du = 1
= \[\int_{-1}^{1}\]c\[u^{2}\]du = 1
=\[\frac{2}{3}\]c
Thus, the value of the constant is \[\frac{2}{3}\]
FAQs on Probability Density Function
1. What is a probability density function in the context of continuous random variables?
A probability density function (PDF) describes the likelihood of a continuous random variable falling within a specific interval. Instead of giving the probability of the variable taking an exact value (which is always zero for continuous variables), the PDF allows calculation of probabilities over an interval using integration. The total area under the curve of the PDF across all possible values equals one.
2. What key properties must a probability density function satisfy according to the CBSE 2025-26 syllabus?
A valid probability density function (PDF) must meet these essential conditions:
- Non-negativity: f(x) ≥ 0 for all x
- Normalization: The integral of f(x) over the entire possible range equals one, i.e., ∫−∞∞ f(x) dx = 1
- Continuity: For a continuous random variable, f(x) is defined and continuous over its domain
3. How is the probability that a continuous random variable lies within a specific interval calculated using its PDF?
To find the probability that a continuous random variable X lies between two values a and b, use:
- P(a < X < b) = ∫ab f(x) dx
Here, f(x) is the probability density function of X.
4. What is the difference between a probability density function (PDF) and a probability mass function (PMF)?
The PDF applies to continuous random variables and is used to calculate probabilities over intervals via integration, while the PMF is used for discrete random variables and assigns probabilities to individual outcomes directly. For continuous variables, the probability at any exact value is zero, but for discrete variables, it can be positive and is given by the PMF.
5. Can the value of a probability density function be greater than 1? If yes, how is probability still valid?
Yes, the value of a PDF at a specific point can be greater than one, as long as the total area under its curve over its domain equals one. For example, the uniform distribution on [0, 0.5] has f(x) = 2 for values in this range. The probability of the random variable being in any particular interval is determined by integrating the PDF over that interval, not by its height alone.
6. Why is the probability at an exact point zero for a continuous random variable?
For continuous random variables, the number of possible values is infinite within any interval. The probability density function measures the likelihood across intervals, so the probability at any specific value is always zero. This reflects that only probabilities over ranges (not points) have nonzero values for continuous distributions.
7. What are the main applications of probability density functions in real-world scenarios?
- Modeling atmospheric or environmental data (like annual pollutant concentrations)
- Describing outcomes in engineering processes (such as combustion in engines)
- Calculating probabilities in statistics and data science
- Assessing risk management in finance and insurance
- Understanding loss probabilities and outcomes in various fields
8. How do probability density functions relate to popular probability distributions such as the normal distribution?
Probability density functions define the shape and properties of probability distributions. For instance, the normal distribution has a characteristic bell-shaped PDF, which is used extensively in statistics, science, and finance because of its natural modeling of random events. The PDF uniquely represents each distribution (like normal, chi-square, binomial) and determines its specific probabilities and statistical properties.
9. What misconceptions do students often have about probability density functions during board exam preparation?
- Confusing the value of the PDF (f(x)) with the probability (which is obtained via integration over an interval)
- Assuming the probability at a specific value of a continuous variable is nonzero
- Not recognising that the area under the curve equals one, regardless of the maximum height of f(x)
- Mixing up PMF and PDF concepts between discrete and continuous variables
10. How can understanding the properties of probability density functions improve your performance in the CBSE Class 12 Maths board exam?
Grasping the meaning and properties of probability density functions enables you to answer conceptual and application-based questions accurately, avoid common mistakes, and solve integration-based probability problems with confidence—skills specifically assessed in the CBSE 2025–26 Mathematics exam pattern.
