# probability mass function vs. cumulative distribution function

One can understand if probability mass function is known then the cumulative distribution function is known and vice-verse. Can someone tell me how they are related to each other?

• What does wherever your "understanding" comes from have to say about the subject? – Dilip Sarwate Nov 18 '14 at 18:12
• Do you see the relation between $P(X\leq x)$ and $P(X=x)$ especially when $X$ is a discrete random variable? That's it. – drhab Nov 18 '14 at 18:44

They are related to each other in the same way as path and velocity, for example. If you plot a graph of velocity depending on time, the the path on segment $[a,b]$ will be equal to an area between that graph and horizontal axis limited by $a$ and $b$ vertical lines. Mathematically it means that path is an antiderivative (integral) of velocity and velocity is a derivative of path. So in this example we have $v(t) = S'(t)$. In probability theory we have CDF as analogue of path and PMF (of PDF in case of continuous distribution) as velocity. As CDF $F(x)$ is defined as $F(x) = P\{X < x\}$, where $X$ is considered random variable, the area between PDF and horizontal axis limited by $a$ and $b$ means probability of the event that $X$ is between $a$ and $b$. So, PDF $f(x)$ is a derivatve of CDF $f(x)$.
In case of PMF instead of PDF we will have stairwise CDF $F(x)$ (it is constant between two adjacent possible values of random variable $X$ and increases by $P\{X=x_i\}$ in each $x_i$ for which such probability is not zero) and PMF will look as set of dots. As CDF is not differentiable and PMF is not integrable, PMF is defined as $f(x) = P\{X = x\}$. It is not defined for such $x$ that $P\{X = 0\}$, so, again, PMF $f(x)$ is just a set of dots and maps possible values $x_i$ of random variable $X$ to their probabilities $P\{X = x_i\}.$
• One thing (small, but important when it comes the characteristics of a CDF): $F(x)=P\{X\leq x\}$ (not $F(x)=P\{X<x\}$ – drhab Nov 18 '14 at 19:08