A **distribution function**, also known as a **cumulative distribution function (CDF)**, is used in probability theory to describe the probability that a random variable \( X \) takes a value less than or equal to a given number \( x \). The general form of a distribution function \( F(x) \) is as follows:

\[
F(x) = P(X \leq x)
\]

This means that for any real number \( x \), \( F(x) \) gives the cumulative probability up to that value. Here's how you would write a distribution function for both discrete and continuous random variables:

### 1. **For a Discrete Random Variable:**
If \( X \) is a discrete random variable, the CDF is the sum of the probabilities for all values of \( X \) less than or equal to \( x \).

\[
F(x) = P(X \leq x) = \sum_{x_i \leq x} P(X = x_i)
\]

Example:
If \( X \) takes values 1, 2, 3 with probabilities \( P(X = 1) = 0.2 \), \( P(X = 2) = 0.5 \), and \( P(X = 3) = 0.3 \), then:

- For \( x = 1 \), \( F(1) = 0.2 \)
- For \( x = 2 \), \( F(2) = 0.2 + 0.5 = 0.7 \)
- For \( x = 3 \), \( F(3) = 0.2 + 0.5 + 0.3 = 1 \)

### 2. **For a Continuous Random Variable:**
If \( X \) is a continuous random variable, the CDF is the integral of the probability density function \( f(x) \), which describes the likelihood of \( X \) falling within an infinitesimal interval around \( x \).

\[
F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \, dt
\]

Example:
For a continuous uniform distribution on the interval [0, 1], the probability density function is:

\[
f(x) = \begin{cases} 
1 & \text{if } 0 \leq x \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]

The CDF would be:

\[
F(x) = \begin{cases}
0 & \text{if } x < 0 \\
x & \text{if } 0 \leq x \leq 1 \\
1 & \text{if } x > 1
\end{cases}
\]

### General Steps to Write a CDF:
1. **Identify the type of random variable**: Is it discrete or continuous?
2. **Find the probability mass function (PMF) for discrete or the probability density function (PDF) for continuous variables.**
3. **For discrete variables, sum the probabilities** up to \( x \). For continuous variables, integrate the PDF from \( -\infty \) to \( x \).
4. **Specify conditions for different values of \( x \)**. For example, the CDF might be 0 for \( x \) less than some lower bound, and 1 for \( x \) greater than some upper bound.

Would you like to see more details or have any questions?

Here are 5 related questions for further exploration:
1. How is the probability density function (PDF) related to the cumulative distribution function (CDF)?
2. Can you explain the concept of a probability mass function (PMF) for discrete variables?
3. What is the difference between a cumulative distribution function (CDF) and a survival function?
4. How do you interpret the CDF of a normal distribution?
5. How does the CDF of a continuous uniform distribution look graphically?

**Tip:** The derivative of a continuous CDF is the probability density function (PDF), and the difference between discrete CDF values gives the probability mass function (PMF).

Learn how to write a distribution function (CDF) for both discrete and continuous random variables with detailed examples. This covers key probability concepts, including the sum of probabilities for discrete variables and integration for continuous variables.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation