Foundations — Conditional expectation

4.9.13 · D1 · Maths › Probability Theory & Statistics › Conditional expectation

Parent note Conditional expectation padhne se pehle, tumhe har symbol apna banana hai jo woh tumpar fire karta hai. Yeh page unhe ek ek karke build karta hai, ek smart 12-saal ke bachche ke starting point se. Koi bhi symbol yahaan define hone se pehle appear nahi karta — words mein aur ek picture se pin kiya hua.

0. Har cheez ke neeche ki picture: sample space

Probability mein sab kuch ek bade box ke andar rehta hai jise sample space kehte hain — un saari cheezOn ka set jo possibly ho sakti hain. Hum ise ek rectangle ki tarah draw karte hain, aur imagine karte hain ki probability ko sand ki tarah uspar chhaante hain: sand ki kul matra exactly $1$ hoti hai.

Topic ko yeh kyun chahiye. Conditional expectation bolne wala hai "box ka sirf kuch hissa rakho." "Box ka kuch hissa" samjhne ke liye tumhe pehle poora box dekhna hoga. Neeche har symbol is rectangle ke kisi region ya kisi sand par ek label hai.

1. Symbols $X$ aur $Y$ — random variables

Ise ek machine ki tarah picture karo: tum ek outcome (ek dot) daalo, aur ek number bahar nikalti hai.

• $X$ = "do dice ka sum" outcome "(3,5)" ko number $8$ mein convert karta hai.
• $Y$ = "pehla die" "(3,5)" ko $3$ mein convert karta hai.

Topic ko yeh kyun chahiye: poora subject hai "machine $Y$ se number jaanna mujhe machine $X$ ke number ke baare mein kya bata deta hai?"

2. Symbol $P(X=x)$ — ek probability distribution

Ab dono ideas combine karo. $\{X=x\}$ shorthand hai event ke liye "machine $X$ ne number $x$ produce kiya" — yaani box ka woh region jahan saare dots jo $x$ par map hote hain rehte hain.

Topic ko yeh kyun chahiye: parent ki definition ki pehli line hi hai $E[X\mid B]={\sum }_{x}xP(X=x\mid B)$. Tumhe $P(X=x)$ ko "ek region par sand" ki tarah sunaai dena chahiye warna woh sum meaningless hai.

3. Symbol $\sum$ — cases par ek sum

Ise ek checklist ki tarah picture karo: $x$ ki har value ke liye ek row, row compute karo, column jodon.

Topic ko yeh kyun chahiye: $\sum$ hi woh tarika hai jisse hum "ek number per case" ko "ek grand total" mein jod te hain. Har expectation formula ek $\sum$ hai.

4. Symbol $E[X]$ — plain expectation

Ab hum show ke star ke un-conditional cousin ko build kar sakte hain.

Yeh idea (aur yeh fact ki $E[aX+bY]=aE[X]+bE[Y]$) Expectation and its linearity mein develop ki gayi hai. Topic ko yeh kyun chahiye: conditional expectation same formula hai, bas box shrink karne ke baad compute kiya gaya.

5. Symbol $P(X=x,Y=y)$ — ek joint probability

Do machines same outcome par chalti hain. Toh hum dono ke baare mein ek saath pooch sakte hain.

Box ko ek grid mein split karke picture karo: columns $Y$ ki values hain, rows $X$ ki values hain. Har cell mein kuch sand hai; joint probability ek cell mein sand hai.

Grid, rows, columns, aur marginals ka subject Joint and marginal distributions hai. Topic ko yeh kyun chahiye: yahi collapse parent note mein Tower Rule derivation ka punchline step hai.

6. Symbol $P(X=x\mid Y=y)$ — conditional probability

Yahaan woh move hai jo sab kuch ka dil hai: $Y$ seekhna box ko shrink karta hai.

$P(Y=y)$ se divide kyun? Kyunki jab tum non-matching outcomes delete karte ho, toh bachne wala sand ab $1$ mein nahi jodta — yeh $P(Y=y)$ mein jodta hai. Divide karna bachne walon ko re-inflate karta hai taki woh phir se $1$ total hoon, unhe proper distribution banate hue.

Yeh Conditional probability mein develop kiya gaya hai, aur saare columns ko wapas jodhna Law of total probability hai. Topic ko yeh kyun chahiye: $E[X]$ formula ke andar plain $P(X=x)$ ki jagah yeh conditional version rakh do aur literally $E[X\mid Y=y]$ mil jata hai.

7. Do functions aur ek arrow: $g(y)$ aur $g(Y)$

Parent jo vocabulary lete hain uska aakhri piece.

Topic ko yeh kyun chahiye: definition ke baad sab kuch — Tower Rule, "jo jaanta hai use bahar nikalna," Conditional variance and Eve's law, aur Martingales mein martingale generalisation — $E[X\mid Y]$ ko ek random variable ki tarah treat karta hai. Yeh miss karo aur notation nonsense lagti hai.

Yeh foundations topic ko kaise feed karte hain

Ise upar-se-neeche padho: box machines ko host karta hai, machines ki distributions hoti hain, distributions expectations mein sum hoti hain, aur conditioning + functions us expectation ko random object $E[X\mid Y]$ mein turn karte hain.

Equipment checklist

Dayi taraf cover karo aur apne aap ko test karo. Agar koi bhi answer tumhe surprise kare, toh parent note kholne se pehle woh section dobara padho.

Connections

• Conditional expectation (woh parent jiske liye yeh page tumhe prepare karta hai)
• Expectation and its linearity
• Conditional probability
• Law of total probability
• Joint and marginal distributions
• Conditional variance and Eve's law
• Wald's identity
• Martingales