Probability Density Function (PDF) and Cumulative Distribution Function (CDF)
1. Probability Density Function (PDF)
The PDF describes how the probability of a continuous random variable \( X \) is distributed.
Definition:
$$ f_X(x) \ge 0, \quad \int_{-\infty}^{\infty} f_X(x) \, dx = 1 $$
Probability in an interval:
$$ P(a \le X \le b) = \int_a^b f_X(x) \, dx $$
Example: Standard normal PDF:
$$ f_X(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2} $$
- Peak at \( x = 0 \)
- Tails approach zero for large \( |x| \)
2. Cumulative Distribution Function (CDF)
The CDF gives the probability that \( X \) is less than or equal to a value:
$$ F_X(x) = P(X \le x) = \int_{-\infty}^{x} f_X(t) \, dt $$
Properties:
- Monotonically non-decreasing
- \( \lim_{x \to -\infty} F_X(x) = 0 \)
- \( \lim_{x \to \infty} F_X(x) = 1 \)
Example: Standard normal CDF:
$$ F_X(x) = 0.5 \cdot \left[1 + \text{erf}\left(\frac{x}{\sqrt{2}}\right)\right] $$
At \( x = 0 \), \( F_X(0) = 0.5 \) → 50% probability below zero.
3. Relationship Between PDF and CDF
$$ F_X(x) = \int_{-\infty}^{x} f_X(t) \, dt $$
$$ f_X(x) = \frac{d}{dx} F_X(x) $$
- PDF → density at a point
- CDF → accumulated probability up to a point
4. Quick Visual Intuition
- PDF: Shape of the curve; peaks indicate likely values
- CDF: Step-like accumulation; always increasing from 0 to 1
5. Summary Table
| Function | Meaning | Formula | Range |
|---|---|---|---|
| PDF \( f_X(x) \) | Probability density | \( f_X(x) \ge 0, \; \int f_X(x) \, dx = 1 \) | \( \ge 0 \) |
| CDF \( F_X(x) \) | Cumulative probability | \( F_X(x) = \int_{-\infty}^{x} f_X(t) \, dt \) | 0 to 1 |