4.9.8
Intuition Bada picture (YE PAANCH KYU?)Continuous distributions un quantities ko describe karti hain jo kisi bhi range mein koi bhi value le sakti hain. Ye paanch "workhorse" hain kyunki ye sab ek alag physical story model karte hain:
Uniform — ek interval ke andar poori ignorance ("har point equally likely").Normal — bahut saare chhote effects ke sums/averages (Central Limit Theorem).Exponential — pehle random event tak ka waiting time (memoryless).Gamma — k-ve event tak ka waiting time (sum of exponentials).Beta — ek random probability / proportion jo [ 0 , 1 ]
Inhe ek family tree ki tarah seekho, paanch alag-alag facts ki tarah nahi. Yahi hai 80/20: stories master karo aur baaki sab algebra hai.
Ek continuous random variable X probability density function f ( x ) ≥ 0 ∫ − ∞ ∞ f ( x ) d x = 1 Cumulative distribution function F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t f ( x ) = F ′ ( x )
Definition Gamma function (hamara integration tool)Γ ( α ) = ∫ 0 ∞ t α − 1 e − t d t , α > 0 on the fly derive karenge: Γ ( α + 1 ) = α Γ ( α ) Γ ( 1 ) = 1 Γ ( n ) = ( n − 1 )! Γ ( 2 1 ) = π
"Mujhe pata hai X a b flat density.
U ( 2 , 8 ) μ = 2 2 + 8 = 5 Kyun? midpoint. Var = 12 ( 8 − 2 ) 2 = 3 P ( X ≤ 5 ) = 8 − 2 5 − 2 = 2 1 Ye step kyun? Uniform ka CDF linear hota hai: b − a x − a
Rate λ pehle event ka wait time. Ye memoryless hota hai: 5 minute pehle se wait kar chuke ho toh bhi future wait ke baare mein kuch nahi pata.
λ = 0.5 = 1/0.5 = 2 P ( wait > 3 ) = e − 0.5 ⋅ 3 = e − 1.5 ≈ 0.223 Ye step kyun? survival = e − λ t
Sirf pehle event ka nahi, α α = k X k Exp ( λ ) Exponential, Gamma hai α = 1 .
Gamma ( 3 , 2 ) = 3/2 = 1.5 = 3/4 = 0.75 Kyun? 3 exponentials ka sum jinka har ek mean 1/2 3 × 2 1 = 1.5 3 × 4 1
Central Limit Theorem : bahut saare independent chhote influences add karo aur total approximately bell-shaped hoga, chahe originals kuch bhi hon. Isliye measurement errors, heights, aur exam scores Normal dikhte hain.
N ( 170 , 1 0 2 ) P ( X ≤ 185 ) z = 10 185 − 170 = 1.5 Ye step kyun? Standard scale par lao. Φ ( 1.5 ) ≈ 0.933 P ( ∣ Z ∣ < 1 ) ≈ 0.68
Ek random number 0 aur 1 ke beech — kisi anjaan probability ya proportion ko model karne ke liye perfect. Shape bahut flexible hai: α = β = 1 α , β > 1 α , β < 1
Beta ( 2 , 5 ) = 7 2 ≈ 0.286 β > α Kyun? zyada "failure mass" average ko left push karta hai.
Common mistake "Exponential rateλ Kyun sahi lagta hai: λ Fix: λ rate hai; mean uska reciprocal 1/ λ
U ( a , b ) ( b − a ) 2 /4 Kyun sahi lagta hai: range b − a Fix: integral se ( b − a ) 2 / 12 3 1 − 4 1
N ( μ , σ 2 ) Kyun sahi lagta hai: bell same shape ki dikhti hai. Fix: tables sirf N ( 0 , 1 ) Hamesha pehle z = ( x − μ ) / σ
Common mistake "Normal ka parameterσ 2 σ 2 σ 2 π Kyun sahi lagta hai: notation N ( μ , σ 2 ) σ 2 Fix: σ 2 π z standard deviation σ
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho popcorn pop hone ka wait. Uniform aise hai jaise "prize is ruler par kahin chhupa hai, koi clue nahi." Exponential hai "pehli pop tak kitna time" — aur iska koi fark nahi kitna pehle se wait kar rahe ho, agli pop utni hi surprise hai. Gamma hai "paanchvi pop tak kitna time" — paanch waits ek ke upar ek. Normal famous bell hai: agar bahut saari random cheezein average karo (jaise bahut saare bacchon ki heights), to average hamesha woh hill shape banata hai. Beta ek dial hai jo sirf 0 se 1 tak jaata hai — bahut useful hai jab jo cheez guess kar rahe ho woh khud ek chance hai, jaise "kitne fraction free throws main karunga?"
Mnemonic Family yaad rakho"U Eat Gamma's Normal Beta" →
U niform (flat), E xponential (1st event, 1/ λ G amma (k α / λ N ormal (bell, CLT), B eta (proportions in [ 0 , 1 ] α / ( α + β ) Exp = Gamma(α = 1 ; sum of Exps = Gamma ; Uniform = Beta(1,1) .
Central Limit Theorem — kyun Normal har jagah aata hai.Poisson Process — Exponential aur Gamma dono generate karta hai.Gamma Function aur Beta Function — integration tools.Moment Generating Functions — means/variances nikalne ka slick tarika.Conjugate Priors in Bayesian Inference — Beta, Binomial ka conjugate hai.Discrete Distributions — Geometric, Exponential ka discrete cousin hai.
Uniform U ( a , b ) f ( x ) = b − a 1 a ≤ x ≤ b
Uniform variance 12 ( b − a ) 2
Exponential PDF and CDF f ( t ) = λ e − λ t F ( t ) = 1 − e − λ t t ≥ 0
Exponential mean and variance mean 1/ λ 1/ λ 2
Memoryless property P ( X > s + t ∣ X > s ) = P ( X > t ) = e − λ t
Relation Exp↔Gamma Exp( λ ) ( α = 1 , λ ) k ( k , λ )
Gamma PDF f ( x ) = Γ ( α ) λ α x α − 1 e − λ x x > 0
Gamma mean and variance mean α / λ α / λ 2
Normal PDF Normal ko standardize karo Z = ( X − μ ) / σ ∼ N ( 0 , 1 ) P ( X ≤ x ) = Φ (( x − μ ) / σ )
68-95-99.7 rule ≈ 68% 1 σ 95% 2 σ 99.7% 3 σ
Beta PDF B ( α , β ) x α − 1 ( 1 − x ) β − 1 ( 0 , 1 ) B = Γ ( α + β ) Γ ( α ) Γ ( β )
Beta mean and variance mean α + β α ( α + β ) 2 ( α + β + 1 ) α β
Beta(1,1) barabar hai Uniform(0,1).
Gamma function recursion Γ ( α + 1 ) = α Γ ( α ) ,
Γ ( n ) = ( n − 1 )! ,
Γ ( 1/2 ) = π .
Normal mein 2 π Gaussian integral
∫ e − z 2 /2 d z = 2 π (polar-coord proof) usse normalize karta hai.
max ignorance flat density
models a random probability
Mean and Variance integrals
Poisson process rate lambda
