Discrete random variables — PMF, CDF — Maths notes

WHAT is a discrete random variable?

WHY do we bother? The raw sample space $\Omega$ (e.g. "the dice landed showing a 3 and a 4") is awkward. We usually care about a number extracted from it (e.g. "the sum is 7"). The RV $X$ is the translator from messy outcomes to numbers we can do algebra on.

WHAT does "discrete" buy us? Because the values are countable, we can assign probability to individual points and just add them. (For continuous RVs each point has probability 0 — there you integrate instead. That contrast is the whole reason this subtopic exists.)

The PMF — Probability Mass Function

HOW must a valid PMF behave? Derive it from the axioms of probability:

Probabilities are never negative, so $p_X(x) \ge 0$ for all $x$ . (Why? Axiom: $P(A)\ge 0$ .)
The events $\{X = x_i\}$ are disjoint (X can't equal both 2 and 5 at once) and together cover everything X can do. By countable additivity: $\sum_i p_X(x_i) = P\!\left(\bigcup_i \{X=x_i\}\right) = P(\Omega) = 1.$

The CDF — Cumulative Distribution Function

HOW is it built from the PMF? You just sum all the mass at points $\le x$ . For a discrete RV this makes $F_X$ a step function: flat between the values, jumping up by exactly $p_X(x_i)$ at each value $x_i$ .

Figure — Discrete random variables — PMF, CDF

Interval probabilities — the payoff

Worked examples

Common mistakes (steel-manned)

Recall Feynman: explain it to a 12-year-old

Imagine a row of buckets numbered 1, 2, 3… and you pour 1 liter of water total, splitting it among the buckets however the game says. The PMF is "how much water is in each bucket." The total is always exactly 1 liter (that's the rule). The CDF is "if I walk from bucket 1 onward and keep adding up the water, how much have I collected by the time I reach this bucket?" When you pass a full bucket your total jumps up; between buckets it stays flat. The last bucket always brings you to a full 1 liter.

Active recall

What makes a random variable discrete?

Its set of possible values is countable (finite or countably infinite), so each value can carry its own positive probability mass.

Define the PMF

p_X(x)

p_X(x) = P(X = x)

, the probability mass on the exact value

x

State the two conditions a valid PMF must satisfy.

p_X(x) \ge 0

for all

x

, and

\sum_x p_X(x) = 1

Define the CDF

F_X(x)

in terms of the PMF.

F_X(x) = P(X \le x) = \sum_{x_i \le x} p_X(x_i)

What shape is a discrete CDF?

A right-continuous step (staircase) function: flat between values, jumping up by

p_X(x_i)

at each value.

How do you recover the PMF from the CDF?

p_X(x_i) = F_X(x_i) - F_X(x_i^-)

— the size of the jump at

x_i

Give the formula for

P(a < X \le b)

F_X(b) - F_X(a)

(excludes

a

, includes

b

What are the limiting values of any CDF?

F_X(-\infty)=0

and

F_X(+\infty)=1

, and it is non-decreasing.

Why is a CDF non-decreasing?

Increasing

x

only adds non-negative probability mass to the cumulative sum; nothing is ever subtracted.

For

p_X(k)=ck

k=1,2,3

, find

c

6c=1 \Rightarrow c=\tfrac16

, from the normalization

\sum p_X=1

Connections

Probability Axioms — countable additivity is what makes $\sum p_X = 1$ .
Continuous random variables — PDF, CDF — same CDF idea, but mass becomes density (integrate, not sum).
Expectation and Variance of Discrete RVs — built directly on the PMF: $E[X]=\sum x\,p_X(x)$ .
Binomial Distribution, Poisson Distribution — named PMFs you'll use constantly.
Conditional Probability — conditional PMFs $p_{X\mid Y}$ reuse this machinery.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek discrete random variable ka matlab bas itna hai ki kisi random experiment ke har outcome ko ek number de do. Jaise do dice phenke, toh "sum" ek random variable hai jiski values 2 se 12 tak ho sakti hain. "Discrete" isliye kyunki values countable hain — alag-alag points, beech mein kuch nahi.

Ab PMF (probability mass function) ka kaam hai batana ki har point pe kitni probability "mass" rakhi hai, yaani $p_X(x)=P(X=x)$ . Do hi rule hain: koi bhi mass negative nahi ho sakti, aur saari masses jodne par total exactly $1$ aana chahiye. Yeh "sum = 1" wala rule normalization karta hai — jab bhi koi constant $c$ nikaalna ho, isi rule se nikalega.

CDF (cumulative distribution function) $F_X(x)=P(X\le x)$ ek running total hai — number line pe left se right chalo aur jitni mass milti jaaye uska total banao. Discrete case mein yeh ek staircase (seedhiyan) banta hai: do values ke beech flat, aur har value pe utna upar jump karta hai jitni us point ki mass hai. Isiliye smooth curve mat banao!

Sabse important practical baat: range ki probability nikaalni ho toh $P(a<X\le b)=F(b)-F(a)$ . Yahan endpoints dhyan se — $<$ aur $\le$ ka farq discrete mein real hota hai kyunki har point ki apni positive mass hoti hai. Exam mein yahi galti sabse zyada hoti hai, toh bracket type hamesha check karo.

Discrete random variables — PMF, CDF

WHAT is a discrete random variable?

The PMF — Probability Mass Function

The CDF — Cumulative Distribution Function

Interval probabilities — the payoff

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Probability Theory & Statistics

Connections