4.9.13
Conditional expectation answer karta hai: "Agar mujhe pehle se kuch information pata hai, toh X
Plain expectation E [ X ] jab kuch bhi nahi pata . Conditional expectation E [ X ∣ Y ] updated guess hai Y — aur sabse important baat, kyunki Y random variable hoti hai.
Definition Kisi event par conditional expectationEk discrete random variable X B P ( B ) > 0 E [ X ∣ B ] = ∑ x x P ( X = x ∣ B ) = ∑ x x P ( B ) P ( X = x , B ) X conditional probability distribution use karke B
Definition Ek variable par conditional expectationE [ X ∣ Y = y ] number hai (har value y E [ X ∣ Y = y ] = ∑ x x P ( X = x ∣ Y = y ) ( discrete ) E [ X ∣ Y = y ] = ∫ − ∞ ∞ x f X ∣ Y ( x ∣ y ) d x ( continuous ) g ( y ) = E [ X ∣ Y = y ] random input Y E [ X ∣ Y ] = g ( Y )
Do flavours kyun? E [ X ∣ B ] fixed event par condition karta hai → ek number. E [ X ∣ Y ] y Y
Yeh THE key theorem hai. Isse scratch se build karte hain.
First principles se derivation (discrete):
E [ X ∣ Y = y ] Y E [ E [ X ∣ Y ] ] = ∑ y E [ X ∣ Y = y ] P ( Y = y )
Yeh step kyun? E [ X ∣ Y ] Y g ( Y ) ∑ y g ( y ) P ( Y = y )
Definition E [ X ∣ Y = y ] = ∑ x x P ( X = x ∣ Y = y ) = ∑ y ( ∑ x x P ( X = x ∣ Y = y ) ) P ( Y = y )
Kyun? Bas plug in kar rahe hain. Ab P ( X = x ∣ Y = y ) P ( Y = y ) = P ( X = x , Y = y ) = ∑ y ∑ x x P ( X = x , Y = y ) = ∑ x x P ( X = x ) y ∑ P ( X = x , Y = y )
Inner sum kyun collapse hoti hai? Saare y y marginalise ho jaata hai, sirf P ( X = x ) = ∑ x x P ( X = x ) = E [ X ] ■
Intuition Tower Rule kyun sach honi chahiyeSocho exam scores average karna. Pehle har class ke andar average karo (E [ X ∣ Y ] Y class size ke hisaab se weight karke (P ( Y = y ) saare students ko ek saath average karo. Split karke phir combine karne se overall mean change nahi ho sakta.
Worked example 1. Do dice roll karo; total ki expected value given pehla even haiMaano X Y E [ X ∣ Y even ] E [ X ∣ Y = y ] = y + E [ doosra die ] = y + 3.5 Kyun? Doosra die independent hai, toh Y Y Y ∈ { 2 , 4 , 6 } 1/3 y = 4 E [ X ∣ Y even ] = 4 + 3.5 = 7.5
Worked example 2. Random number of summands (Wald-style)Ek fair coin flip karo; heads par X = X = Y E [ X ∣ Y = H ] = 3.5 E [ X ∣ Y = T ] = 7 Tower: E [ X ] = 2 1 ( 3.5 ) + 2 1 ( 7 ) = 5.25 Yeh step kyun? Hum kabhi X
Worked example 3. Continuous:Y ∼ U ( 0 , 1 ) X ∣ Y = y ∼ U ( 0 , y ) E [ X ∣ Y = y ] = 2 0 + y = 2 y [ 0 , y ] E [ X ∣ Y ] = Y /2 Tower: E [ X ] = E [ Y /2 ] = 2 1 E [ Y ] = 2 1 ⋅ 2 1 = 4 1 Kyun? Hum joint density par mushkil direct integral ko do aasaan steps se replace karte hain.
E [ X ∣ Y ] Kyun sahi lagta hai: E [ X ∣ Y = y ] hota hai ek number, toh log distinction drop kar dete hain.
Fix: E [ X ∣ Y = y ] y E [ X ∣ Y ] Y random variable hai — yehi reason hai ki E [ E [ X ∣ Y ] ]
E [ X Y ∣ Y ] = Y E [ X ] Kyun sahi lagta hai: "known stuff nikaal bahar" aur "independence E [ X ] Fix: Y E [ X Y ∣ Y ] = Y E [ X ∣ Y ] E [ X ∣ Y ] E [ X ] independence chahiye. Dono rules ko fuse mat karo.
Common mistake Tower meinP ( Y = y ) Kyun sahi lagta hai: conditional means ko average karna symmetric lagta hai.
Fix: E [ X ] = ∑ y E [ X ∣ Y = y ] P ( Y = y )
Recall Feynman: 12-saal ke bacche ko samjhaoTum guess karna chahte ho ki school ka koi random baccha kitna tall hai. Tera pehla guess school ka average hai — woh hai E [ X ] Y 8th-grade average par — woh hai E [ X ∣ Y ]
"Averages ko average karo, crowds ke hisaab se weight karke." — Tower Rule.
Aur: E andar, E bahar, beech wala gayab → E [ E [ X ∣ Y ]] = E [ X ]
E [ X ∣ Y ] Y Ek random variable — random Y g ( Y )
E [ X ∣ Y = y ] Ek number, har fixed value y
Tower / Law of Iterated Expectation state karo. E [ E [ X ∣ Y ] ] = E [ X ]
Tower derive karne ka pehla step (discrete)? E [ E [ X ∣ Y ]] = ∑ y E [ X ∣ Y = y ] P ( Y = y )
Inner sum ∑ y P ( X = x , Y = y ) y P ( X = x )
"Take out what is known" rule kya hai? E [ g ( Y ) X ∣ Y ] = g ( Y ) E [ X ∣ Y ]
Agar X ⊥ Y E [ X ∣ Y ] Constant E [ X ]
Y ∼ U ( 0 , 1 ) X ∣ Y = y ∼ U ( 0 , y ) E [ X ] E [ X ∣ Y ] = Y /2 E [ X ] = E [ Y /2 ] = 1/4
Coin: H→ek die, T→do dice ka sum. E [ X ] 2 1 ( 3.5 ) + 2 1 ( 7 ) = 5.25
Tower groups par apply karte waqt common slip kya hai? Conditional means ko P ( Y = y )
random Y ko g of Y mein daalo
averaging se wapas milta hai
g of Y constant ki tarah act karta hai
E X given Y is a random variable
Tower Rule E of E X given Y equals E X
