Q.Distinguish between the probability mass function (p.m.f.) and the cumulative distribution function (c.d.f.) of a discrete random variable. State two properties that any c.d.f. must satisfy.

The p.m.f. gives the probability of each individual value of a random variable, while the c.d.f. gives the accumulated probability of all values up to and including a given point — the two are related by addition/subtraction.

Probability mass function (p.m.f.): for a discrete random variable $X$, $p(x_i) = P(X=x_i)$ gives the probability that $X$ takes exactly the value $x_i$. It answers questions of the form "what is the chance $X$ equals this particular value?"

Cumulative distribution function (c.d.f.): $F(x) = P(X\leq x)$ gives the probability that $X$ takes a value up to and including $x$. It is obtained from the p.m.f. by cumulative addition, $F(x) = \sum_{x_i\leq x} p_i$, and it answers questions of the form "what is the chance $X$ is at most this value?"

Relationship: the two can always be converted into each other — the c.d.f. is the running total of the p.m.f., and conversely the p.m.f. at a value $x_i$ can be recovered from the c.d.f. as the jump at that point: $p(x_i) = F(x_i) - F(x_i^{-})$, where $F(x_i^{-})$ is the c.d.f. value just before $x_i$.

Two properties every valid c.d.f. must satisfy:

1. $F(x)$ is non-decreasing — it never falls as $x$ increases, since accumulated probability can only stay the same or grow.
2. $F(x) \to 0$ as $x \to -\infty$ and $F(x) \to 1$ as $x \to \infty$ — before the smallest possible value, nothing has been accumulated yet; by the largest possible value, everything has.

(A third useful property, not asked for above but often stated alongside these: $0\leq F(x)\leq1$ for every $x$, since $F(x)$ is itself a probability.)

Students often confuse the p.m.f. value $p(x_i)$ with the c.d.f. value $F(x_i)$ at the same point — they are equal only for the very first (smallest) value of $X$; for every later value, $F(x_i)$ also includes all the smaller values' probabilities, so $F(x_i) \geq p(x_i)$ always, with equality only at the first point.