PDF vs. CDF: Difference Between PDF and CDF

PDF describes the probability distribution of a continuous random variable, while PDF describes the probability distribution of both discrete and continuous random variables.

Discover the key difference between PDF and CDF. Understand how they shape data distribution and drive statistical analysis. Uncover the pivotal variance between these fundamental statistical concepts.

The probability distribution of a random variable is a list of all possible outcomes with corresponding probability values, and different functions are used to describe this probability distribution. Probability Density Function and Cumulative Distribution Function are among such functions.

The probability Density Function describes the probability distribution of continuous random variables. The Cumulative Distribution Function deals with continuous and discrete random variables.

In this article, we will learn the difference between PDF and PDF in probability.

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Before starting the article, let’s discuss random variables.

A Random Variable is a numerical description of the outcome of a random experiment. Generally, it is represented by X.

It is of two types: 

Discrete Random Variable: It has a countable number of values between two values.

Example: Number of Heads 

Continuous Random Variable: It has an infinite number of values between two values.

Example: Distance Travelled

Now, let’s start with the probability density function vs cumulative distribution function based on the different parameters.

Table of Content

• PDF vs CDF
• What is Probability Density Function (PDF)
• What is Cumulative Distribution Function (CDF)
• Relation between PDF and CDF
• Key Difference between PDF and CDF

CDF vs PDF: Difference Between PDF and CDF

What is the Probability Density Function

A probability density function (PDF) is a fundamental concept in probability theory and statistics. It is used to describe the probability distribution of a continuous random variable. The PDF represents the likelihood of the random variable taking on a specific value or falling within a particular range of values. Mathematically, the PDF is a function that, when integrated over a range, gives the probability that the random variable falls within that range. It is a crucial tool for understanding and analyzing continuous probability distributions in various scientific and engineering fields.

• It tells the probability that a random variable takes on a certain value. 

Must Read: Probability Density Function

Formulas

The Probability density function of a continuous random variable X is a function f_X: R -> [0, inf) such that:

Where R is a real number.

Properties

• The value of PDF is always greater than or equal to 1.
• The area under the PDF curve is always equal to 1.
• The median divides the PDF curve into two equal halves.

Example

If

f(x) = x, if 0 < x < 1

f(x) = 2 – x, if 1 < x < 2 and

f(x) = 0, otherwise

Then, f(x) is a probability density function centred at x = 1.

What is the Cumulative Distribution Function?

A cumulative distribution function (CDF) is a function that gives the probability that a random variable is less than or equal to a certain value. In other words, it provides the cumulative probability of the random variable taking on a value less than or equal to a given value. The CDF is an essential concept in probability theory and statistics and is widely used to analyze and understand the behaviour of random variables.

Formula

The CDF (Cumulative Distribution Function) of a random variable X is defined as

F_X(X) = P (X <= X),

for all x in R.

Properties

• CDF is a non-decreasing.
• The minimum of CDF is 0 when x = – infinity: Fx(- inf) = 0.
• The maximum of CDF is 1 when x = infinity: Fx(inf) = 1
• P [a <= x <= b] = F_X(b) – F_X(a)

Example

Consider the pdf function given in the above example and find the CDF for ½ <= x <= 3/2.

Answer:

Relation Between PDF and CDF

• The probability distribution function (PDF) gives the probability that a random variable takes on a specific value.
• The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a certain value.
• The CDF is the integral of the PDF, which means the CDF accumulates the probabilities from the PDF up to a certain point.
• The PDF and CDF are related such that the CDF can be obtained by integrating the PDF, and the PDF can be obtained by differentiating the CDF.
• The CDF provides a complete picture of the probabilities associated with a random variable, while the PDF focuses on the probability density at specific points.

What are the key differences between PDF and CDF?

1.  The probability distribution function (PDF) gives the probability that a random variable takes on a specific value, while the cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a certain value.
2. The PDF provides the probability density at specific points, while the CDF accumulates the probabilities up to a certain point.
3. The CDF is always non-decreasing, as it represents the accumulation of probabilities, whereas the PDF does not have this restriction.
4. The CDF ranges from 0 to 1, while the PDF can take any non-negative value.
5. The CDF can be obtained by integrating the PDF, and the PDF can be obtained by differentiating the CDF.

Conclusion

In this article, we have discussed the probability density function (PDF) and cumulative density function (CDF), their properties and the difference between CDF and PDF with the help of examples.

PDF (or Probability Density Function) is a function that describes the probability of a continuous random variable. The PDF function’s curve looks like a bell-shaped curve, taking the value between the given intervals. In contrast, CDF or Cumulative Distribution Function is a probability that a random variable takes on a value less than or equal to X. The CDF can be defined. for both the random variables: continuous and discrete random variables.

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