1. Meaning of the Cumulative Distribution Function
The Cumulative Distribution Function (CDF) of a random variable X gives the probability that X takes a value less than or equal to x. It is written as:
\( F(x) = P(X \le x) \)
The CDF adds up all probabilities up to the point x. It applies to both discrete and continuous random variables.
2. How the CDF Works
For any value of x, the CDF shows how much of the total probability has been accumulated up to that point.
Intuitively, it tells us how far along the probability has been collected when we move from left to right on the number line.
2.1. Example Idea
If X represents the number on a die, then:
- \( F(1) = P(X \le 1) = 1/6 \)
- \( F(3) = P(X \le 3) = 3/6 = 1/2 \)
- \( F(6) = P(X \le 6) = 1 \)
The CDF climbs step by step as x increases.
3. Properties of a CDF
A proper CDF must satisfy important properties:
- Non-decreasing: It never goes down. As x increases, F(x) stays the same or increases.
- Right-continuous: There are no jumps when coming from the right side of any point.
- Limits at the ends:
\( \lim_{x \to -\infty} F(x) = 0 \)
\( \lim_{x \to +\infty} F(x) = 1 \)
4. CDF of a Discrete Random Variable
For discrete variables, the CDF increases in jumps at the values where the PMF has probability. Between these values, the CDF stays constant.
4.1. Example: Die Roll
Let X be the number shown on the die. The CDF values are:
| x | F(x) |
|---|---|
| 1 | 1/6 |
| 2 | 2/6 |
| 3 | 3/6 |
| 4 | 4/6 |
| 5 | 5/6 |
| 6 | 1 |
This CDF jumps at each integer value from 1 to 6.
5. CDF of a Continuous Random Variable
For continuous variables, the CDF is smooth and continuous. It is obtained by integrating the PDF.
\( F(x) = \int_{-\infty}^{x} f(t)\,dt \)
The CDF increases continuously as more area is collected under the PDF curve.
5.1. Example: Uniform Distribution on [0, 1]
If the PDF is:
\( f(x) = 1 \quad \text{for } 0 \le x \le 1 \)
Then the CDF is:
\( F(x) = 0 \quad (x < 0) \)
\( F(x) = x \quad (0 \le x \le 1) \)
\( F(x) = 1 \quad (x > 1) \)
The CDF rises steadily from 0 to 1 as x moves from 0 to 1.