Independence of random variables — formal definition: Foundations

4.9.11 · D1 · Maths › Probability Theory & Statistics › Independence of random variables — formal definition

Is page par yeh assume kiya gaya hai ki aapne parent note ki notation pehle kabhi nahi dekhi. Hum har symbol ko ground up se build karte hain, ek aisi order mein jahan har ek sirf pichle waalon par depend karta hai.

0. Shuruaati picture: outcomes aur probability

Ek fair experiment imagine karo — ek spinning wheel, ek rolled die, ek measured height. Jab bhi aap ise run karte ho, ek cheez hoti hai. Woh "ek cheez jo hoti hai" ko outcome kehte hain.

Topic ko yeh kyun chahiye: independence probabilities ke baare mein ek statement hai, isliye hume object $P$ ki zaroorat hai jo woh numbers deta hai — baaki sab se pehle.

1. Events aur symbol $\cap$

Symbol $\cap$ ka matlab hai "aur" — do events ka overlap.

Figure — Independence of random variables — formal definition

(Figure s01) — Do event-circles $A$ (lavender) aur $B$ (coral). Yeh picture symbol $\cap$ ko concrete banati hai: shaded lens jahan circles overlap karti hain wahi $A \cap B$ hai, woh region jo $P$ measure karta hai jab hum poochte hain "dono hote hain".

Topic ko yeh kyun chahiye: events ke liye yeh multiply-rule seed hai. Independent variables ka poora idea yahi hai: is rule ko har us sawaal ke liye sach banao jo aap pooch sakte ho. Dekho Independence of events.

2. Random variable $X$ — outcomes se numbers tak

(Figure s02) — Teen-box flow outcome → X → number literally dikhata hai ki symbol $X$ kya karta hai: ek die-outcome butter-coloured machine $X$ mein enter hota hai aur ek bare number bahar nikalta hai. Isliye ${X \in A}$ niche ek event hai — yeh un saare outcomes ko name karta hai jinke exiting number $A$ mein land hote hain.

Notation ${X \in A}$ (jahan $A$ ab numbers ka ek set hai, jaise interval $(- \infty, x]$ ) khud ek event hai: "woh number jo $X$ produce karta hai woh $A$ ke andar land karta hai."

Topic ko yeh kyun chahiye: variables ki independence is demand se define hoti hai ki ${X \in A}$ aur ${Y \in B}$ saare number-sets $A, B$ ke liye independent events hon.

3. Distribution, CDF, PMF, PDF — ek variable ki bhaashayein

Ek random variable ki poori personality uski distribution hai: uske output-numbers kaise spread out hote hain. Iske teen languages hain.

(Figure s03) — Ek hi variable teen languages mein side by side: lavender CDF $0 \to 1$ tak chadhti hai, coral PMF discrete spikes ke roop mein, aur mint PDF jahan area ka ek shaded slice ek probability hai. Teeno ko ek picture se padhna hi aapko allow karta hai ki parent ke CDF-, PMF- aur PDF-factorisation rules ko ek hi idea ke roop mein pehchano.

Topic ko yeh kyun chahiye: parent ke teen criteria (CDF, PMF, PDF factorisation) in teen languages mein likha hua wahi ek statement hai. Teeno ko pehchaanna zaroori hai.

4. Ek saath do variables: joint aur marginal

Ab show ka star. Jab ek hi experiment par do machines $X$ aur $Y$ chal rahe hain, toh ek outcome ek pair $(X, Y)$ produce karta hai — plane mein ek point.

(Figure s04) — Lavender cloud joint pair $(X, Y)$ hai. Har point ko seedha $x$ -axis par neeche dabaana (coral) $Y$ ko bhool jaata hai aur $X$ ka marginal chhodta hai; baayein $y$ -axis par dabaana (mint) $Y$ ka marginal chhodta hai. Yeh do shadows exactly wahi hain jo neeche wale marginal integrals/sums compute karte hain.

5. Symbols $⊥$ , product, aur "support"

(Figure s05) — Baayein, ek mint rectangle support: har horizontal slice same range of $x$ offer karta hai, isliye $y$ fix karna $x$ ke baare mein kuch naya nahi batata. Daayein, ek coral triangle support: ek bada $x$ choose karna chhota $y$ forbid karta hai. Yeh picture isliye hai kyunki "support ek box hona chahiye" ek genuine extra condition hai, decoration nahi.

Topic ko yeh kyun chahiye: parent ke "consequences" section mein dikhaya gaya hai ki independence $Cov = 0$ force karti hai, aur warn karta hai ki reverse fail hota hai. Ise padhne se pehle aapko yeh chahiye. Dekho Covariance and correlation.

6. Yeh sab topic ko kaise feed karta hai

Equipment checklist

Outcome kya hota hai, aur sample space

Ω

kya hai?

Ek outcome experiment ka ek single result hota hai;

Ω

saare possible outcomes ka set hai.

A \cap B

ka kya matlab hai, ek picture mein?

Overlap ("dono hote hain") — woh lens jahan do event-circles cross karti hain.

Event-independence rule state karo.

P (A \cap B) = P (A) P (B)

: dono ka chance, chances ka product hai.

Random variable kya hota hai?

Ek machine jo ek outcome read karke ek number return karti hai.

{X \in A}

ek event kyun hai?

Yeh yes/no sawaal hai "kya

X

ka number set

A

mein land kiya?", isliye saare event-rules apply hote hain.

CDF

F_{X}

define karo.

F_{X} (x) = P (X \leq x)

, chance ki output zyaada se zyaada

x

ho; yeh

0

1

tak chadhta hai.

PMF vs PDF — kaun spikes use karta hai, kaun area?

PMF

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

spike-heights hain (discrete); PDF

f_{X}

density hai jahan probability curve ke neeche area hoti hai (continuous).

Discretely aur continuously, joint se marginal kaise nikalte hain?

Discrete:

p_{X} (x) = \sum_{y} p_{X, Y} (x, y)

; continuous:

f_{X} (x) = \int f_{X, Y} (x, y) d y

(doosre variable ko collapse karo).

CDF, PMF aur PDF languages mein independence criterion state karo.

F_{X, Y} = F_{X} F_{Y}

;

p_{X, Y} = p_{X} p_{Y}

;

f_{X, Y} = f_{X} f_{Y}

— joint marginals ke product ke barabar hoti hai.

X ⊥ Y

ka kya matlab hai aur iske saath kaun si surface-picture jaati hai?

X, Y

independent hain; joint object do marginal shadows ke product ke barabar hai.

Discrete aur continuous variables ke liye support kya hota hai?

(x, y)

ka woh set jahan pair land kar sakta hai: jahan

p_{X, Y} > 0

(discrete spikes) ya

f_{X, Y} > 0

(continuous density).

Factorisation test ke liye support ka rectangle hona kyun zaroori hai?

Warna ek variable ke allowed values doosre par depend karte hain (info leak hoti hai), jo independence tod deta hai chahe formula split ho.

Covariance formula do aur woh kya detect karta hai.

Cov (X, Y) = E [X Y] - E [X] E [Y]

; yeh linear co-movement detect karta hai (

0

= koi linear tendency nahi).

Connections

Independence of Random Variables — Formal Definition
Independence of events
Joint distribution and marginals
Covariance and correlation
Conditional distributions and conditional independence