CDF vs. PDF

Difference Between Distribution Function and Probability Distribution Function

1. Distribution Function (CDF)

Definition:
The distribution function (also called the Cumulative Distribution Function, CDF) is defined as:

F_X(x) = P(X ≤ x)

It gives the probability that a random variable takes a value less than or equal to x.

Properties:

• Non-decreasing function
• 0 ≤ F_X(x) ≤ 1
• F_X(−∞) = 0
• F_X(+∞) = 1

2. Probability Distribution Function

The probability distribution function describes how probability is distributed over values of a random variable. It depends on whether the random variable is discrete or continuous.

(a) Discrete Random Variable — PMF

Definition:
P(X = x_i) = p(x_i)

Example:

X = {0, 1, 2}
P(X = 0) = 0.2,  P(X = 1) = 0.5,  P(X = 2) = 0.3

This table represents the probability mass function (PMF).

(b) Continuous Random Variable — PDF

Definition:
A probability density function f_X(x) satisfies:

P(a ≤ X ≤ b) = ∫_a^b f_X(x) dx

Example:

Uniform distribution:

f_X(x) =
1/2,  0 ≤ x ≤ 2
0,  otherwise

3. Relationship Between CDF and PDF / PMF

Continuous Case

F_X(x) = ∫_−∞^x f_X(t) dt

f_X(x) = dF_X(x) / dx

Discrete Case

F_X(x) = ∑ p(x_i),  for all x_i ≤ x

4. Side-by-Side Example

Continuous Example

Given: f_X(x) = 2x,  0 ≤ x ≤ 1

CDF:
F_X(x) = 0,  x < 0
F_X(x) = x²,  0 ≤ x ≤ 1
F_X(x) = 1,  x > 1

Discrete Example

X	1	2	3
P(X)	0.2	0.3	0.5

CDF values:
F_X(1) = 0.2
F_X(2) = 0.5
F_X(3) = 1

5. Key Differences

Aspect	Distribution Function (CDF)	Probability Distribution Function
Meaning	P(X ≤ x)	PMF / PDF
Type	Cumulative	Point-wise / Density
Values	0 to 1	PDF can exceed 1
Nature	Non-decreasing	Describes shape
Use	Find probabilities	Model randomness

Summary

• Use CDF to find probabilities over intervals
• Differentiate CDF to get PDF
• Probability at a point for continuous RV is zero