Cumulative Distribution Function

What’s CDF?

The Cumulative Distribution Function (CDF) represents the probability that a random variable $X$ takes a value less than or equal to $x$: $F(x)=P(X\le x)$

Benefits of the CDF

1. Probability Calculation: Easily compute probabilities for intervals: $P(a\le X\le b)=F(b)-F(a)$
2. Describes Distribution: Provides a complete picture of the random variable’s behavior.
3. Quantiles & Percentiles: Solve $F(x)=p$ to find critical values.
4. Comparison Tool: Compare distributions visually and analytically.
5. Applications:
   • Risk Management: Estimate probabilities of extreme losses.
   • Reliability Engineering: Determine failure probabilities.
   • Finance & Machine Learning: Model distributions and analyze performance.

For Discrete

If $X$ is a discrete random variable: $F(x)={\sum }_{t=-\infty }^{x}P(t)\text{where }P(t)\text{ is the p.m.f.}$

For Continuous

If $X$ is a continuous random variable: $F(x)={\int }_{-\infty }^{x}f(t)dt\text{where }f(t)\text{ is the p.d.f.}$

Properties of CDF

1. $0\le F(x)\le 1$ for all $x$.
2. $F(x)$ is non-decreasing.
3. ${\lim }_{x\to -\infty }F(x)=0$ and ${\lim }_{x\to \infty }F(x)=1$.
4. If $F(x)$ is continuous and differentiable, then the p.d.f. is the derivative of the CDF: $f(x)=\frac{dF(x)}{dx}$

Map

• Probability and Statistics Lecture 8

References

• Lectures at MPSTME

Information

• date: 2025.03.13
• time: 08:10