Central Limit Theorem — statement, proof sketch, significance

4.9.15 · Maths › Probability Theory & Statistics

THEOREM kya kehta hai

HAR piece ka matlab:

$\overset{ˉ}{X}_{n} - μ$ : sample average truth se kitna door hai.
$σ / n$ se divide karna: $\overset{ˉ}{X}_{n}$ ka standard deviation exactly $σ / n$ hota hai (yeh standard error hai), toh hum rescale kar rahe hain taaki spread hamesha $1$ ho.
Limit $X_{i}$ ki shape ki parwah nahi karta — sirf yahi chahiye ki mean aur variance finite hon.

Figure — Central Limit Theorem — statement, proof sketch, significance

PROOF kaise karte hain (characteristic-function sketch)

Step 1 — Center aur scale karo. Define karo $Y_{i} = \frac{X _{i} - μ}{σ}$ . Toh $E [Y_{i}] = 0$ , $Var (Y_{i}) = 1$ , aur $Z_{n} = \frac{1}{n} \sum_{i = 1}^{n} Y_{i} .$ Yeh step kyun? Isse $μ, σ$ remove ho jaate hain taaki hum sirf standardized case prove kar sakein; algebra cleaner ho jaati hai.

Step 2 — Independence use karke CF ko factor karo. $φ_{Z_{n}} (t) = E [exp (i t \frac{1}{n} \sum Y_{i})] = \prod_{i = 1}^{n} E [e^{i (t / n) Y_{i}}] = [φ_{Y} (\frac{t}{n})]^{n} .$ Yeh step kyun? Independence ⇒ product ki expectation = expectations ka product; identical distribution ⇒ sab factors equal hain, isse $n$ -th power milta hai.

Step 3 — CF ko $0$ ke paas Taylor-expand karo. Kyunki $E [Y] = 0$ , $E [Y^{2}] = 1$ , $φ_{Y} (s) = 1 + i s E [Y] - \frac{s ^{2}}{2} E [Y^{2}] + o (s^{2}) = 1 - \frac{s ^{2}}{2} + o (s^{2}) .$ Yeh step kyun? Pehle do moments hi woh data hain jo CLT use karta hai; baaki sab $n \to \infty$ par vanish ho jaate hain. $s = t / n$ rakhte hain: $φ_{Y} (\frac{t}{n}) = 1 - \frac{t ^{2}}{2 n} + o (\frac{1}{n}) .$

Step 4 — $n$ -th power ki limit lo. $φ_{Z_{n}} (t) = (1 - \frac{t ^{2}}{2 n} + o (\frac{1}{n}))^{n} n \to \infty e^{- t^{2} /2} .$ Yeh step kyun? Classic limit $(1 + \frac{a _{n}}{n})^{n} \to e^{a}$ jab $a_{n} \to a$ , yahan $a = - t^{2} /2$ hai.

Step 5 — Limit ko pehchano. $e^{- t^{2} /2}$ exactly $N (0, 1)$ ki CF hai. Lévy's continuity theorem se, $Z_{n} d N (0, 1)$ . $■$

Recall Proof ko 5 words mein reconstruct karo

Standardize. 2) Independence factors CF. 3) Taylor to second moment. 4) $(1 - \frac{t ^{2}}{2 n})^{n} \to e^{- t^{2} /2}$ . 5) That's $N (0, 1)$ 's CF.

$n$ kyun, $n$ kyun nahi? (key scaling)

Worked examples

Common mistakes (steel-manned)

Significance (YEH kyun matter karta hai)

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho school ke har bacche ne ek jar mein candies ki number guess ki. Ek bacche ki guess wild hai — bahut door. Lekin agar aap pure school ki guesses ka average lo, toh jo guesses bahut zyada thi aur jo bahut kam thi woh cancel ho jaati hain, aur average sachchi number ke paas pahunch jaata hai. Central Limit Theorem ek aur magical baat kehta hai: agar aap bahut saare alag-alag schools imagine karo jo apna-apna average banayein, toh woh averages, jab plot karo, hamesha ek smooth hill shape banate hain — "bell" — chahe individual bacche bilkul crazy tarike se guess karein. Aur jitne zyada bacche per school, utni hi patli aur reliable woh hill hoti hai.

Flashcards

Classical CLT

X_{i}

se kya require karta hai?

Independent, identically distributed, with finite mean

μ

and finite variance

σ^{2} > 0

\overset{ˉ}{X}_{n}

ka standard error kya hai?

σ / n

Z_{n} = (\overset{ˉ}{X}_{n} - μ) / (σ / n)

ki CLT limit distribution kya hai?

N (0, 1)

Specifically

n

se divide kyun karte hain?

Kyunki

SD (\sum X_{i}) = σ n

; yeh normalization non-degenerate limit deta hai (

n

se divide karo → 0, LLN; 1 se → ∞).

Kaun sa tool "independent vars ka sum" ko product mein badalta hai?

Characteristic function

φ_{X} (t) = E [e^{i tX}]

(CFs independent sums ke liye multiply hoti hain).

0

ke paas standardized CF ka Taylor expansion?

φ_{Y} (s) = 1 - \frac{s ^{2}}{2} + o (s^{2})

(kyunki

E [Y] = 0, E [Y^{2}] = 1

Proof finish karne wali limit identity kaun si hai?

(1 - \frac{t ^{2}}{2 n})^{n} \to e^{- t^{2} /2}

, jo

N (0, 1)

ki CF hai.

Kaun sa theorem CF convergence se distributional convergence imply karta hai?

Lévy's continuity theorem.

Kya CLT raw data ko normal banata hai?

Nahi — sirf mean/sum ki sampling distribution ko.

Classical CLT kab fail hota hai?

Jab variance infinite ho (jaise Cauchy distribution).

P (X \leq k)

X

discrete ke liye continuity correction?

Φ ((k + 0.5 - μ) / σ)

use karo.

Standard error aadha karne ke liye kitna zyada data chahiye?

4 \times

(error

\propto 1/ n

\sum_{i = 1}^{n} X_{i}

ka mean aur variance?

n μ

aur

n σ^{2}

Bernoulli sums ke liye CLT ka discrete special case ka naam batao.

de Moivre–Laplace theorem (binomial ka normal approximation).

Connections

Law of Large Numbers — same setup, lekin average ki value converge hoti hai; CLT uske fluctuations describe karta hai.
Characteristic Functions — proof ka engine.
Normal Distribution — universal limit.
Confidence Intervals aur Hypothesis Testing — z and t tests — direct applications.
Standard Error — $σ / n$ scaling.
Binomial Distribution — de Moivre–Laplace as a CLT instance.
Lindeberg–Feller Condition — non-identical, independent terms ka generalization.
Cauchy Distribution — famous counterexample (finite variance nahi hai).