Common distributions (Bernoulli, Binomial, Poisson)

1.3.10 · AI-ML › Probability & Statistics

Overview

Ye teen discrete probability distributions alag-alag types ke counting experiments ko model karti hain. Inhe samajhna classification, A/B testing, rare event prediction, aur ML mein Bayesian inference ke liye bahut zaroori hai.

1. Bernoulli Distribution

PMF (Probability Mass Function): $P (X = x) = p^{x} (1 - p)^{1 - x}, x \in {0, 1}$

Ye form kyun? Jab $x = 1$ ho, toh hume milta hai $p^{1} \cdot (1 - p)^{0} = p$ . Jab $x = 0$ ho, toh milta hai $p^{0} \cdot (1 - p)^{1} = 1 - p$ . Exponents sahi probability "select" kar lete hain.

2. Binomial Distribution

PMF: $P (X = k) = (k n) p^{k} (1 - p)^{n - k}, k = 0, 1, 2, \dots, n$

First principles se derivation:

$p^{k} (1 - p)^{n - k}$ kyun? Hume exactly $k$ successes aur $n - k$ failures chahiye. Agar trials independent hain, toh probability = (individual probabilities ka product).
$(k n)$ kyun? $k$ successes kisi bhi order mein aa sakte hain. $n$ mein se $k$ positions choose karne ke tarike hain $(k n) = \frac{n !}{k ! ( n - k )!}$ . Har arrangement ki same probability $p^{k} (1 - p)^{n - k}$ hai, isliye hum count se multiply karte hain.

3. Poisson Distribution

PMF: $P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}, k = 0, 1, 2, \dots$

Binomial ki limit ke roop mein derivation:

Socho $n$ bahut bada hai, $p$ bahut chhota hai, lekin $n p = λ$ (constant average rate).

Binomial se shuru karo: $P (X = k) = (k n) p^{k} (1 - p)^{n - k}$

$p = λ / n$ substitute karo: $P (X = k) = \frac{n !}{k ! ( n - k )!} (\frac{λ}{n})^{k} (1 - \frac{λ}{n})^{n - k}$

Simplify karo: $= \frac{n ( n - 1 ) \dots ( n - k + 1 )}{n ^{k}} \cdot \frac{λ ^{k}}{k !} \cdot (1 - \frac{λ}{n})^{n} \cdot (1 - \frac{λ}{n})^{- k}$

Jab $n \to \infty$ :

$\frac{n ( n - 1 ) \dots ( n - k + 1 )}{n ^{k}} \to 1$ (leading terms dominate karte hain)
$(1 - \frac{λ}{n})^{n} \to e^{- λ}$ ( $e$ ki definition)
$(1 - \frac{λ}{n})^{- k} \to 1$

Result: $P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}$

Ye sensible kyun lagta hai? Poisson Binomial hai "infinite trials, infinitesimal success probability, finite expected count" ke saath. Rare events ke liye perfect.

Distributions Ke Beech Connections

Figure — Common distributions (Bernoulli, Binomial, Poisson)

Distribution	Kab Use Karein	Parameters	Mean	Variance
Bernoulli	Single binary trial	$p$	$p$	$p (1 - p)$
Binomial	Fixed # of trials	$n, p$	$n p$	$n p (1 - p)$
Poisson	Rare events, continuous opportunities	$λ$	$λ$	$λ$

Key insight: Binomial → Poisson jab $n \to \infty$ , $p \to 0$ , $n p = λ$ constant ho.

Common Mistakes

ML Applications

Logistic Regression output: Bernoulli likelihood (binary labels ke liye $P (y = 1∣ x)$ model karo).
A/B Testing: Conversion counts ke liye Binomial models. $p_{A}$ vs $p_{B}$ compare karo.
Anomaly Detection: Rare events ke liye Poisson (fraud, system failures). Agar observed count >> $λ$ , toh anomaly flag karo.
Count Data Regression: Counts predict karne ke liye Poisson regression (# of purchases, # of bugs).
Naive Bayes with binary features: Har feature class ke conditional pe Bernoulli hai.

Recall Feynman Explanation (ELI12)

Socho tumhare paas ek magic coin hai. Jab tum ise ek baar flip karte ho, ya toh heads (success) aata hai ya tails (failure). Ye hai Bernoulli – ek flip, do outcomes.

Ab wahi coin 10 baar flip karo aur count karo kitne heads aaye. Ye hai Binomial – multiple flips, wins count karo. Math batata hai: "Agar mera coin 50-50 hai, toh 10 mein se exactly 7 heads aane ke kitne chances hain?" Formula mein do parts hain: ek kehta hai "is exact pattern (HHHHTT) ke kya chances hain?" aur doosra kehta hai "10 flips mein 7 heads kitne alag-alag tareekon se arrange ho sakte hain?"

Poisson super rare cheezoon ke liye hai. Jaise, "1000-page book mein kitne typos hain?" Tum 1000 coins flip nahi karte—har word mein typo ho sakta hai, lekin ye bahut unlikely hai. Poisson shortcut hai jab kisi rare cheez ke liye bazillion chances hon, aur tum sirf jaanna chahte ho "kitne dikhenge?" Cool part ye hai: average aur "spreadiness" same number hote hain!

Teeno counting ke baare mein hain, lekin sahi wala choose karo: ek flip = Bernoulli, fixed flips = Binomial, rare stuff jo bade space/time mein ho = Poisson.

[!mnemonic] BP Memory Hook "Bern-ONE, Bi-MANY, Poi-RARE"