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:

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).