4.9.6
Intuition Badi picture (YE PAANCH KYU EXIST KARTE HAIN)Probability mein almost har "count" wala question basically ek hi cheez ke paanch roop hai — ek single trial : ek coin flip jisme success probability p
Bernoulli — coin ko ek baar flip karo. (sabka atom)Binomial — n fixed baar flip karo, successes gino.Geometric — pehli success aane tak flip karo, trials gino.Negative Binomial — r Poisson — Binomial ban jaata hai yeh jab n → ∞ p → 0 n p = λ
Sirf Bernoulli atom + counting argument seekh lo, baki charon apne aap nikal aate hain. Yahi is poore topic ka 80/20 hai.
( p ) X ∈ { 0 , 1 } P ( X = 1 ) = p , P ( X = 0 ) = 1 − p = q . trial ko model karta hai do outcomes ke saath: "success" (1) aur "failure" (0).
Intuition "Choose" kyun aata haiAgar main n k koi bhi specific sequence ki probability p k q n − k k ( k n ) kaun se trials succeed hue. Unhe jod do.
( n , p ) X = n ( p ) P ( X = k ) = ( k n ) p k q n − k , k = 0 , 1 , … , n .
Worked example Kharab bulbs10 bulbs hain, har ek p = 0.1 P ( X = 2 ) = ( 2 10 ) ( 0.1 ) 2 ( 0.9 ) 8 . Yeh step kyun? ( 2 10 ) = 45 ( 0.1 ) 2 ( 0.9 ) 8 = 45 ⋅ 0.01 ⋅ 0.4305 ≈ 0.194.
q k − 1 p k k − 1 sab fail hone chahiye (q k − 1 k p
( p ) X = P ( X = k ) = q k − 1 p , k = 1 , 2 , 3 , …
Intuition Memorylessness — iska unique fingerprintGeometric akela discrete distribution hai jisme P ( X > m + n ∣ X > m ) = P ( X > n )
Definition Negative Binomial( r , p ) X = r P ( X = k ) = ( r − 1 k − 1 ) p r q k − r , k = r , r + 1 , …
( r − 1 k − 1 ) ( r k ) k zaroor success hona chahiye (woh r stopping event hai). Isliye hum sirf baaki r − 1 k − 1 ( r − 1 k − 1 ) p r r q k − r r = 1
Intuition KAHAN SE AATA HAIBinomial( n , p ) n → ∞ p → 0 expected count n p = λ
Worked example Call centreAverage λ = 3 P ( X = 5 ) = 5 ! 3 5 e − 3 = 120 243 e − 3 ≈ 0.101. λ = 3 λ
Dist.
P ( X = k ) Mean
Variance
Counts
Bernoulli( p )
p k q 1 − k k ∈ { 0 , 1 } p pq 1 trial
Binomial( n , p )
( k n ) p k q n − k n p n pq n
Geometric( p )
q k − 1 p 1/ p q / p 2 pehli success tak trials
Neg.Bin.( r , p )
( r − 1 k − 1 ) p r q k − r r / p r q / p 2 r
Poisson( λ )
λ k e − λ / k ! λ λ rare events / unit
Common mistake Common galtiyan (steel-manned)(a) Negative Binomial ke liye ( r k ) Kyun sahi lagta hai: "Main k r Fix: aakhiri trial forced hai ki stopping success ho, isliye sirf r − 1 k − 1 ( r − 1 k − 1 ) (b) Independence ke bina variances add karna. Kyun sahi lagta hai: means freely add hote hain, toh variances "bhi hone chahiye." Fix: Var ( X + Y ) = Var X + Var Y + 2 Cov ( X , Y ) tabhi zero hota hai jab independent ho.
(c) Poisson ko galat window ke saath use karna. Kyun sahi lagta hai: rate number andhon ki tarah pakad lete ho. Fix: λ λ = 6 (d) Geometric mein off-by-one. Do conventions hain: "trials ki sankhya" (k ≥ 1 1/ p k ≥ 0 q / p
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek thoda tedha coin flip kar rahe ho jo kabhi kabhi heads (=jeet) aata hai.
Bernoulli: ek flip — jeet gaye? Haan/nahi.Binomial: theek 10 baar flip karo — kitni baar jeete?Geometric: tab tak flip karo jab tak pehli jeet na aaye — kitna waqt laga?Negative Binomial: tab tak flip karo jab tak 5vi jeet na aaye.Poisson: flips ki jagah, ek ghante mein kitne shooting stars dekhte ho jab har second chance bahut chhoti ho lekin raat lambi ho.
Wahi coin, alag sawaal . Bas yahi hain paanchon distributions.
Mnemonic BIG-PN ordering — "kab rukta hoon?"B ernoulli (1 flip) → B inomial (fixed n G eometric (1st success pe ruko) → N egBin (r P oisson (kabhi mat ruko — continuous time, rare hits).
Memory line: "Be In Good Numbers, Please" — Bernoulli, Binomial, Geometric, NegBin, Poisson.
Bernoulli mean and variance E [ X ] = p Var = pq X 2 = X
Binomial PMF mein ( k n ) k p k q n − k ( k n )
Binomial mean and variance n p n pq n
Geometric PMF (trials convention) P ( X = k ) = q k − 1 p k ≥ 1
Geometric mean 1/ p ∑ k x k − 1 = 1/ ( 1 − x ) 2
Kaun sa distribution memoryless hai? Geometric (discrete) — akela woh jisme P ( X > m + n ∣ X > m ) = P ( X > n )
Negative Binomial PMF ( r − 1 k − 1 ) p r q k − r k ≥ r
NegBin ke liye ( r − 1 k − 1 ) ( r k ) k r r − 1 k − 1
NegBin mean and variance r / p r q / p 2 r
Poisson PMF λ k e − λ / k ! k ≥ 0
Poisson kis cheez ka limit hai? Binomial( n , p ) n → ∞ p → 0 n p = λ
Poisson mean and variance Dono λ
Poisson ki key signature mean = variance.
Geometric, NegBin hai jisme kaun sa r r = 1
Variances kab add kar sakte ho? Tabhi jab variables independent hon (covariance term zero ho jaata hai).
Bernoulli trial — paanchon ka shared atomExpectation and Variance — yahan har formula inhi definitions se derive hota haiLinearity of expectation — Binomial/NegBin means turant deta haiGeometric series aur uska derivative — Geometric mean ka engineLimit e^x as (1+x/n)^n — Poisson derivation ka enginePoisson process — Poisson distribution ka continuous-time originMemorylessness — Geometric (discrete) aur Exponential distribution (continuous) ko jodata haiMoment generating functions — in sab moments tak ek alternative unified rasta
Bernoulli p - single trial
