1. Let's start by stating the problem: What is a distribution function? 2. A distribution function, often called a cumulative distribution function (CDF), describes the probability that a random variable $X$ takes a value less than or equal to a certain value $x$. 3. The formula for a distribution function $F(x)$ is: $$F(x) = P(X \leq x)$$ This means $F(x)$ gives the probability that $X$ is at most $x$. 4. Important properties of distribution functions: - $F(x)$ is non-decreasing: as $x$ increases, $F(x)$ does not decrease. - $\\lim_{x \to -\infty} F(x) = 0$ and $\\lim_{x \to +\infty} F(x) = 1$. - $F(x)$ is right-continuous. 5. In simple terms, the distribution function accumulates probabilities from the left up to $x$, showing how likely it is for the variable to be less than or equal to $x$. 6. For example, if $X$ is a discrete random variable with values $1, 2, 3$ and probabilities $0.2, 0.5, 0.3$ respectively, then: $$F(1) = P(X \leq 1) = 0.2$$ $$F(2) = P(X \leq 2) = 0.2 + 0.5 = 0.7$$ $$F(3) = P(X \leq 3) = 0.2 + 0.5 + 0.3 = 1$$ This shows how probabilities accumulate. 7. Distribution functions are fundamental in probability and statistics to understand the behavior of random variables.