Foundations — Monte Carlo simulation — law of large numbers basis

4.9.25 · D1 · Maths › Probability Theory & Statistics › Monte Carlo simulation — law of large numbers basis

Is page pe kuch bhi assume nahi kiya gaya. Parent note mein har ek squiggle — $\mathbb{E}$, $\mu$, ${\sigma }^2$, ${\bar{X}}_n$, $\sum$, $1_A$, $\stackrel{P}{\to }$ — yahan ek picture se build ki gayi hai, use karne se pehle. Upar se neeche padho; har block agla block earn karta hai.

0. "Random variable" $X$ kya hota hai?

Simple shabdon mein. Ek random variable bas ek aisa number hai jo tum abhi nahi jaante kyunki woh kisi random experiment ke outcome pe depend karta hai. Die roll karo → upar wala number ek random variable hai. Ek square pe dart phenko → kya woh circle ke andar gira (haan/nahi, likh lo 1/0) — yeh bhi ek random variable hai.

Picture. Ek lever wali machine socho. Lever khicho aur ek number bahar aata hai. Tum predict nahi kar sakte kaun sa — lekin agar bahut baar khicho toh tum pattern zaroor predict kar sakte ho.

Topic ko yeh kyun chahiye. Jo bhi Monte Carlo estimate karta hai ($\pi$, ek integral) — sab kuch "kisi machine se girane wali typical value" ke roop mein rewrite ho jaata hai. Hum us value ko letter $X$ lete hain (ya $g(X)$ jab hum $X$ ko pehle function $g$ se pass karte hain).

1. Uniform machine aur indicator $1$

Parent note mein do special machines baar baar aati hain.

Picture. Indicator ek light switch jaisa hai jo ek rule se bandha hai. Region ke andar giro → light ON (1). Bahar giro → light OFF (0). Iska average exactly woh fraction hai jitna time light on rehti hai — jo ek probability hai. Yeh akela fact $\pi$ estimate karne ka engine hai.

2. Summation sign $\sum$ aur sample mean ${\bar{X}}_n$

Topic ko yeh kyun chahiye. ${\bar{X}}_n$ hi Monte Carlo estimator hai. Page pe baaki sab kuch iss baare mein hai ki ${\bar{X}}_n$ truth ke kitna paas aata hai aur kitni tezi se.

3. Expectation $\mathbb{E}[X]$ aur mean $\mu$ — "true average"

Ab woh crucial distinction jo poora subject ispe ghoomta hai.

Picture. Ek million pulls ka histogram socho. Uska ek balance point hota hai — woh jagah jahan uske neeche ruler rakh do toh perfectly balance ho jaaye. Wahi balance point hai $\mu =\mathbb{E}[X]$.

4. Variance ${\sigma }^2$ — machine kitna wobble karti hai

Squared kyun? Hum distance $X-\mu$ ko square karte hain taaki mean se neeche hona (negative gap) aur upar hona (positive gap) dono spread count karein instead of cancel hone ke. Deep dive: Variance and Covariance.

Picture. Chhota ${\sigma }^2$ = ek tall thin pile jo $\mu$ ke paas hug kar rahi hai (consistent machine). Bada ${\sigma }^2$ = ek wide flat pile (erratic machine). Yeh "width" exactly wahi hai jo zyada samples average karne par shrink hoti hai — yeh secret hai ki Monte Carlo kyun kaam karta hai.

5. "i.i.d." — woh independence jo tum baar baar sunte ho

Picture. $n$ identical machines, har ek ko alag blindfolded insaan ne alag kamre mein ek baar khiincha. Same design (identical), koi communication nahi (independent).

Topic ko yeh kyun chahiye. Identical hone se har $X_i$ ka same $\mu$ hota hai isliye unka average $\mu$ ko target karta hai. Independent hona hi woh cheez hai jo variance ko simply add hone deta hai (§4). Independence todo — correlated samples use karo — aur clean ${\sigma }^2/n$ formula collapse ho jaata hai. Iss spread ko guarantee mein baadalne wali bound ke liye dekho Chebyshev's Inequality.

6. Convergence arrow $\stackrel{P}{\to }$ aur limit

Picture. $\mu$ horizontal line ke around $\epsilon$ half-width ka ek narrow band draw karo. Jaise $n$ badhta hai, ${\bar{X}}_n$ ka wobbling path us band mein ghus jaata hai aur (almost) kabhi bahar nahi nikalta. Is promise ka ek zyada strong version hai Strong Law of Large Numbers.

7. $\theta$, $g(X)$ aur ${\hat{\theta }}_n$ — kisi bhi target ko average ke roop mein dress up karna

8. Standard error ${\sigma }_g/\sqrt{n}$ — "$\sqrt{n}$" kyun aata hai

Ek baar sample mean ka variance ${\sigma }_g^2/n$ ho (§4), toh uski standard deviation square root hai: $\text{SE}=\sqrt{\frac{{\sigma }_g^2}{n}}=\frac{{\sigma }_g}{\sqrt{n}}.$

 ​ kyun? Variance squared units mein hota hai (§4); answer ke same units mein spread paane ke liye square root lete hain, jo $n$ ko bhi root ke neeche khiinch laata hai. Yahi famous "$1/\sqrt{n}$" ki origin hai: error ko $10\times$ shrink karne ke liye $n$ ko $100\times$ badhana padta hai. Error ki bell curve ki precise shape — aur hence confidence intervals — Central Limit Theorem se aati hai.

Prerequisite map

Equipment checklist

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