Discrete random variables — PMF, CDF: Foundations

4.9.3 · D1 · Maths › Probability Theory & Statistics › Discrete random variables — PMF, CDF

Yeh page kuch bhi assume nahi karta. Parent note padhne se pehle, hum har woh symbol build karte hain jo woh silently use karta hai. Upar se neeche padho — har block ek eent hai jis par agla tika hai. PMF symbol $p_{X}$ aur CDF symbol $F_{X}$ ko hum formally §7–§9 mein introduce karte hain; tab tak hum unhe sirf words mein name karte hain.

1. Random experiment aur uske outcomes

Imagine karo. Ek die roll karo. Outcomes chhe faces hain. Har face ek $ω$ hai.

Kyun zaroorat hai. Probability "kya ho sakta hai?" se shuru hoti hai. Numbers ya chances attach karne se pehle, hum chahte hain "jo ek cheez hui" ka koi naam ho. Woh naam hai $ω$ .

Figure — Discrete random variables — PMF, CDF

2. Sample space $Ω$

Imagine karo. Ek box. Box ke andar chhe dice faces baithey hain. Box hai $Ω$ ; andar har dot hai $ω$ .

Symbol $\in$ padha jaata hai "is an element of" / "ke andar hai." Toh $3 \in {1, 2, 3}$ true hai.

3. Sets, events, aur tools ${\dots}$ , $\cup$ , disjoint

${X = x}$ ka matlab hai "un sabhi outcomes $ω$ ka set" jinhein number $X$ value $x$ deta hai." Isliye yeh ek event hai — $Ω$ ka ek subset.
Disjoint sets mein koi common member nahi hota — jaise do aisi boxes jinhein kuch common nahi. Do circles ka picture socho jo overlap nahi karte.
$\cup$ padho "union" — sets ko ek bade set mein glue karta hai: ${1, 2} \cup {3} = {1, 2, 3}$ . Socho do boxes ko ek mein pour karna.

Topic ko yeh kyun chahiye. Parent claim karta hai ${X = 2}$ aur ${X = 5}$ disjoint hain (ek number 2 aur 5 dono nahi ho sakta), aur un sabki union over all values poora $Ω$ hai. Woh ek sentence hi masses ko 1 tak sum karne par force karta hai — toh tumhe "event," $\cup$ , aur "disjoint" ke peeche ka picture dekhna chahiye.

4. Probability $P (\cdot)$ aur uske teen rules

Parent PMF properties ko teen facts (Probability Axioms) se "derive" karta hai. Simple words mein:

Kabhi negative nahi: $P (A) \geq 0$ . Chance kuch se kam nahi ho sakta.
Total ek hai: $P (Ω) = 1$ . Box mein se kuch zaroor hoga.
Disjoint pieces jodo: agar events overlap nahi karte, toh "inme se koi bhi" ka chance unke chances ka sum hai. Yahi countable additivity hai — yeh infinite list of pieces ke liye bhi kaam karta hai.

5. Jinmein map karte hain woh numbers: $R$ aur functions $X : Ω \to R$

Imagine karo. Box $Ω$ mein har dot se ek arrow nikalta hai aur number line $R$ par kisi jagah land karta hai. Arrows ka woh bundle hi random variable $X$ hai.

Topic ko yeh kyun chahiye. Raw outcomes ("die woh face dikhata hai jis par chhe pips hain") ke saath compute karna mushkil hai. $X$ har outcome ko ek number mein translate karta hai taaki hum arithmetic, sum, aur average kar sakein.

6. Countable — woh word jo ise "discrete" banata hai

7. PMF symbol $p_{X}$ aur summation $\sum$

Ab hum pehle bookkeeping tool ka naam le sakte hain.

$all x \sum p_{X} (x)$ ka matlab hai " $p_{X} (x)$ ko un sabhi values $x$ par add karo jo $X$ le sakta hai."
$x_{i} \leq x \sum p_{X} (x_{i})$ ka matlab hai "sirf woh masses jodo jinkei value $x_{i}$ , $x$ se zyada nahi hai."

Topic ko yeh kyun chahiye. Normalization rule ( $\sum p_{X} = 1$ ) aur §8–§9 mein jo running total hum banate hain, dono $\sum$ se likhe jaate hain. Agar $\sum$ opaque hai, toh central formulas unreadable hain — isliye hum ise yahan pin karte hain.

8. CDF symbol $F_{X}$ aur $x_{i}^{-}$ trick

Topic ko yeh kyun chahiye. Discrete RVs ke liye ek single point real chance carry karta hai, toh ek endpoint include ya exclude karne se answer badal jaata hai. Formula $p_{X} (x_{i}) = F_{X} (x_{i}) - F_{X} (x_{i}^{-})$ literally $x_{i}$ par jump ki height measure karta hai — exactly wahan baitha mass. Socho ek seeenhi par khade ho: tumhari height ( $F_{X} (x_{i})$ ) minus thoda left ki height ( $F_{X} (x_{i}^{-})$ ) barabar step ka rise ( $p_{X} (x_{i})$ ).

9. Vocabulary ko saath jodte hain (yeh parent hai, miniature mein)

Ab $p_{X} (x) = P (X = x)$ aur $F_{X} (x) = \sum_{x_{i} \leq x} p_{X} (x_{i})$ mein har symbol ka matlab hai:

Symbol	Simple words	Picture
$ω, Ω$	ek outcome, saare outcomes ka box	dots in a box
event / subset of $Ω$	outcomes ka ek chunk	box ka ek region
$X : Ω \to R$	har outcome ko number tag karne wala rule	line par arrows
${X = x}$	event: value $x$ dene waale outcomes	box ka ek slice
$P (\cdot)$	ek event ka chance, $[0, 1]$ mein	kitna paani
countable	listable value set	numbered buckets
$p_{X} (x)$	PMF: value $x$ par mass	ek bar ki height
$\sum$	add up karo	buckets saath pour karo
$\leq, x_{i}^{-}$	include / just-left-of	seenhi ka endpoint
$F_{X} (x)$	CDF: mass ka running total	kitni height chadhe

Parent jo kuch karta hai — masses, staircases, interval formulas — yahi eentein stack hain.

Prerequisite map

Equipment checklist

Sawaal padho, zor se jawab do, phir reveal karo. Agar koi trip kare, toh woh section dobara padho.

ω

symbol kya represent karta hai,

Ω

ke mukable?

ω

ek akela outcome hai;

Ω

saare possible outcomes ka set hai (poora sample space).

Zor se padho:

ω \in Ω

"Outcome omega, sample space Omega ka ek element hai (ke andar hai)."

Event kya hota hai?

Sample space

Ω

ka ek subset — outcomes ka ek chunk jise hum probability assign kar sakein.

\cup

kya karta hai, aur "disjoint" ka matlab kya hai?

\cup

(union) sets ko ek mein merge karta hai; disjoint ka matlab sets mein koi common member nahi — koi overlap nahi.

Teen probability axioms ko simple words mein batao.

P (A) \geq 0

(kabhi negative nahi);

P (Ω) = 1

(kuch zaroor hoga); disjoint events apni probabilities add karte hain (countable additivity).

X : Ω \to R

ka matlab kya hai?

X

ek rule hai jo har outcome leta hai aur ek real number return karta hai — outcomes se numbers ka ek translator.

"Countable" ek variable ko discrete kyun banata hai?

Countable values listable hain, isliye har value ka apna positive chunk of probability ho sakta hai jise hum add karte hain — integration ki zaroorat nahi.

PMF symbol

p_{X} (x)

ka matlab kya hai?

p_{X} (x) = P (X = x)

, value

x

par exactly baitha probability mass.

\sum_{i = 1}^{3} i

evaluate karo.

1 + 2 + 3 = 6

CDF symbol

F_{X} (x)

ka matlab kya hai?

F_{X} (x) = P (X \leq x) = \sum_{x_{i} \leq x} p_{X} (x_{i})

x

tak collect hue mass ka running total.

\leq

aur

<

mein kya difference hai, aur yahan yeh kyun matter karta hai?

\leq

endpoint include karta hai,

<

exclude karta hai; discrete RVs ke liye ek point real mass rakhta hai, toh isse probability badal jaati hai.

F_{X} (x_{i}^{-})

ka matlab kya hai, aur

p_{X} (x_{i}) = F_{X} (x_{i}) - F_{X} (x_{i}^{-})

kyun?

F_{X} (x_{i}^{-})

x_{i}

ke thoda left tak cumulative total hai; ise

F_{X} (x_{i})

se ghatane par jump height milti hai, jo

x_{i}

par mass hai.

Connections

Discrete random variables — PMF, CDF — woh parent jiske liye yeh page tayaar karta hai.
Probability Axioms — $P (\cdot)$ ke teen rules jo upar use hue.
Continuous random variables — PDF, CDF — "countable" kya rule out karta hai: wahan hum integrate karte hain.
Conditional Probability — same $Ω$ , events, aur $P (\cdot)$ vocabulary reuse karta hai.