Cumulative Distribution Function ( CDF) with Examples in Statistics

In statistics, a Cumulative Distribution Function (CDF) is a function that shows the cumulative probability of a random variable taking on a value less than or equal to a given value. The CDF is defined for both discrete and continuous random variables.

Definition and Properties:

For a random variable $X$ , the CDF $F(x)$ is defined as: $F(x) = P(X \leq x)$ where $P(X \leq x)$ denotes the probability that $X$ is less than or equal to $x$ .

Key properties of the CDF:

• $F(x)$ is non-decreasing: As $x$ increases, $F(x)$ either increases or stays constant.
• $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$ : The CDF approaches 0 as $x$ goes to negative infinity and approaches 1 as $x$ goes to positive infinity.
• $F(x)$ is right-continuous: $F(x)$ does not jump; it changes smoothly or remains constant.

Examples:

Example 1: Discrete Random Variable

Let's consider a discrete random variable $X$ that represents the outcome of rolling a fair six-sided die.

• Possible outcomes: $\{1, 2, 3, 4, 5, 6\}$
• Probability mass function (PMF): $$P(X = x) = \frac{1}{6}, \quad \text{for } x = 1, 2, 3, 4, 5, 6$$

The CDF $F(x)$ for this discrete random variable is: $F(x) = P(X \leq x)$

For example,

• $F(2) = P(X \leq 2) = P(X = 1) + P(X = 2) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}$

Example 2: Continuous Random Variable

Consider a continuous random variable $Y$ that follows a uniform distribution on the interval $[0, 1]$ .

• Probability density function (PDF): $$f(y) = 1, \quad \text{for } 0 \leq y \leq 1$$

The CDF $F(y)$ for this continuous random variable is: $F(y) = P(Y \leq y) = \int_{-\infty}^{y} f(t) \, dt$

For example,

• $F(0.5) = P(Y \leq 0.5) = \int_{0}^{0.5} 1 \, dt = 0.5$

Uses of CDF:

• Probability Calculation: $F(x)$ can be used to find probabilities for both discrete and continuous distributions.
• Median and Percentiles: The median is the value of $x$ such that $F(x) = 0.5$ . Percentiles can be found similarly.
• Comparison of Distributions: CDFs are useful for comparing different distributions or assessing the fit of a distribution to data.

In summary, the Cumulative Distribution Function (CDF) in statistics is a fundamental concept that provides a comprehensive view of the probability distribution of a random variable, both for discrete and continuous cases. It allows us to calculate probabilities, find medians and percentiles, and compare distributions effectively.