Probability and Statistics Lecture 8

Continued

Probability and Statistics Lecture 7

Cumulative Distribution Function

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

Link to original

Questions based on Cumulative Distributions

Question 1

Example

Suppose $X$ is a discrete random variable with the following probability distribution:

$X$	1	2	3
$P(X=x)$	1/4	1/2	1/4

Problem

Obtain the cumulative distribution function (CDF) of $X$.

Solution

The distribution function of $X$ is given by: $F(x)=P(X\le x)={\sum }_{x_i\le x}P(x_i)\text{ if }X\text{ is discrete.}$

Therefore:

$$F(x)=\{\begin{array}{ll}0,& x<1\\ \frac{1}{4},& 1\le x<2\\ \frac{3}{4},& 2\le x<3\\ 1,& x\ge 3\end{array}$$

Question 2

Example

Suppose $X$ is a continuous random variable with the probability density function (p.d.f.): $f(x)=\frac{1}{2},0\le x\le 2.$

Problem

Obtain the cumulative distribution function (CDF) of $X$.

Solution

The cumulative distribution function of $X$ is given by: $F(x)=P(X\le x)={\int }_{-\infty }^{x}f(t)dt\text{ if }X\text{ is continuous.}$

For $f(x)=\frac{1}{2},0\le x\le 2$:

$$F(x)={\int }_0^x\frac{1}{2}dt=\frac{1}{2}{|}_0^x=\frac{1}{2}x$$

Final CDF:

$$F(x)=\{\begin{array}{ll}0,& x<0\\ \frac{1}{2}x,& 0\le x\le 2\\ 1,& x\ge 2\end{array}$$

Question 3

References

Continued

Probability and Statistics Lecture 9