A cumulative density function (”CDF”) expresses the probability that the [[Random Variable]] $X$ is less or equal than some value $x$. Both discrete and continuous distributions have a CDF. Below we show that it contains all probabilistic information that is needed to derive a PMF or PDF from it. $ F_X(x)= \mathbf P(X \le x) $ ![[cumulative-density-function.png|center|400]] ## Discrete Distribution We obtain the CDF by summing up the [[Probability Mass Function|PMF]] over all $x$ that are $\le k$. $ F_X(x)= \mathbf P(X \le x) = \sum_{k \le x}p_X(k) $ To get the value of a PMF at $k$ from a CDF, a subtraction by the CDF from the previous $(k-1)$ term is needed. $ p_X(k) = F_X(k)-F_X(k-1) $ ## Continuous Distribution The CDF is the integral of the PDF from the left (i.e. $-\infty$) up to some value $x$. $ F_X(x) = \mathbf P (X \le x) = \int_{-\infty}^x f_X(x) \,dx $ The derivative of an integral where the upper limit is the variable itself, results in the integrant $f_X(x)$. Therefore the derivative of the CDF is the PDF. $ \frac{dF_X(x)}{dx}(x)=f_X(x) $ ## Properties of a Cumulative Density Function - Non decreasing: If some value $b \ge a$ then also $F_X(b) \ge F_X(a)$ - $F_X(x) \to 1$ when $x \to \infty$ - $F_X(x) \to 0$ when $x \to (-\infty)$