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:
FX(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 ≤ FX(x) ≤ 1
- FX(−∞) = 0
- FX(+∞) = 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 = xi) = p(xi)
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 fX(x) satisfies:
P(a ≤ X ≤ b) = ∫ab fX(x) dx
Example:
Uniform distribution:
fX(x) =
1/2, 0 ≤ x ≤ 2
0, otherwise
3. Relationship Between CDF and PDF / PMF
Continuous Case
FX(x) = ∫−∞x fX(t) dt
fX(x) = dFX(x) / dx
Discrete Case
FX(x) = ∑ p(xi), for all xi ≤ x
4. Side-by-Side Example
Continuous Example
Given: fX(x) = 2x, 0 ≤ x ≤ 1
CDF:
FX(x) = 0, x < 0
FX(x) = x2, 0 ≤ x ≤ 1
FX(x) = 1, x > 1
Discrete Example
| X | 1 | 2 | 3 |
|---|---|---|---|
| P(X) | 0.2 | 0.3 | 0.5 |
CDF values:
FX(1) = 0.2
FX(2) = 0.5
FX(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