Foundations — Transformations of random variables — change-of-variable technique

4.9.14 · D1 · Maths › Probability Theory & Statistics › Transformations of random variables — change-of-variable tec

Is page mein assume kiya gaya hai ki parent note ki notation tumne kabhi nahi dekhi. Hum har symbol build karenge, use ek picture se anchor karenge, aur tabhi agla symbol uspe lean karega.

1. Random variable $X$ — ek number jo kisi random process se nikalti hai

Picture. Ek machine ki tarah socho jo har baar button dabane par ek number ugalti hai — kabhi $0.3$, kabhi $0.9$, kabhi $0.51$. Machine woh random process hai; jo number woh print kare woh $X$ hai.

Topic ko iski zaroorat kyun hai. Poora subject yahi hai: "Mujhe pata hai ek machine ke numbers kaise behave karte hain; main un numbers ko ek function mein feed karta hoon; nayi machine kaise behave karegi?" Yeh sawaal hi nahi puch sakte bina number ka naam liye: $X$.

• Capital $X$ = machine / woh number pehle se, jab nahi pata konsi value aayi.
• Lowercase $x$ = ek specific value jo machine print kar sakti hai (number line par ek fixed point).

In dono ko alag rakhna sabse zyada kaam aane wali aadat hai. $P(X\le x)$ padha jaata hai "woh chance ki machine kuch print kare jo fixed spot $x$ par ya usse neeche ho."

2. Number line aur support — paint kahan land kar sakti hai

Picture (upar wali figure). Ek horizontal ruler. Shaded band woh jagah hai jahan values possible hain; uske bahar machine kabhi nahi print karti, toh wahan zero paint hai.

Topic ko iski zaroorat kyun hai. Axis bend karne ke baad, nayi variable sirf wahan land kar sakti hai jahan purani wali kar sakti thi. Agar $X$ $(0,1)$ par rehta hai aur $Y=X^2$ hai, toh $Y$ sirf $(0,1)$ mein ho sakta hai. Support bhool jaana parent note ke chaar classic mistakes mein se ek hai.

3. Probability density $f_X$ — paint kitni thick hai

Picture (upar wali figure). Ek curve ruler ke upar baitha hua hai. Kisi point $x$ par uski height $f_X(x)$ hai. Jahan curve zyada tall hai, paint thick hai (wahan values zyada likely hain); jahan low hai, paint thin hai.

"Per unit length" kyun matter karta hai. Akeli height ka koi matlab nahi — amount of paint paane ke liye tumhe width se multiply karna hoga. Yahi agla idea hai aur poori technique ka dil hai.

4. Differential $dx$ — ek infinitely thin slab of width

Picture. Ruler ko zoom karo jab tak slab $[x,x+dx]$ itna narrow na ho jaaye ki density curve uske upar flat dikhe. Uska area ek thin rectangle hai: $\text{(slab par paint ki matra)}=\underset{\text{height}}{\underbrace{f_X(x)}}\times \underset{\text{width}}{\underbrace{dx}}.$

Topic ko iski zaroorat kyun hai. Poora change-of-variable equation hai: $f_Y(y)|dy|=f_X(x)|dx|,$ yaani "after-slab par paint ki matra = before-slab par matra." Us line ka har symbol ab ek picture rakhta hai.

5. Function $Y=g(X)$ — ruler ko bend karna

Picture (upar wali figure). Do parallel rulers: neeche wala $x$-axis hai, upar wala $y$-axis hai. Curve $g$ neeche ke har point ko upar ke ek point tak map karta hai. Neeche width $dx$ ka ek slab upar width $dy$ ke slab mein carry ho jaata hai — aur yeh usually ek alag width hoti hai. Width mein yahi farq exactly woh reason hai ki paint thickness badlti hai.

• Monotonic = curve sirf upar (increasing) ya sirf neeche (decreasing) jaata hai; kabhi palatta nahi. Toh har $y$ exactly ek $x$ se aata hai.
• Non-monotonic = curve palat jaata hai (jaise $g(x)=x^2$, ek valley), toh ek $y$ do alag $x$ values se aa sakta hai.

6. Inverse $g^{-1}$ — map ko backwards padhna

Picture. Two-ruler figure par, upar ek point $y$ chuno aur wapas neeche arrow pe slide karo woh $x$ dhundne ke liye jis se woh aaya. Yahi backward slide $g^{-1}$ hai.

Topic ko iski zaroorat kyun hai. Final formula $y$ ke terms mein likha jaata hai (hum $Y$ ke baare mein jaanna chahte hain), lekin $f_X$ sirf $x$ ke baare mein jaanta hai. "Woh $x$ jo is $y$ ko produce karta hai" par $f_X$ evaluate karne ke liye, hum us $x$ ko $g^{-1}(y)$ likhna hoga. Monotonic $g$ exactly ek aisa $x$ guarantee karta hai, yahi wajah hai ki monotonic wala clean case hai.

7. Derivative $\frac{dx}{dy}$ — stretch factor (the Jacobian)

Picture (upar wali figure). Neeche ruler par width $dx$ ka before-slab aur upar width $dy$ ka after-slab jismein woh map hota hai. Agar after-slab double as wide hai, toh uski paint half as thick hai. Ratio $\left|\frac{dx}{dy}\right|$ tumhe exactly batata hai ki thickness kitni rescale hoti hai.

Absolute value kyun. Widths lengths hain — hamesha positive. Agar $g$ decrease karta hai, $y$ mein aage badhna matlab $x$ mein peeche badhna, toh $\frac{dx}{dy}$ negative aata hai. Ise $|\cdot |$ mein wrap karna woh direction phenk deta hai aur sirf stretch ka size rakhta hai, toh density positive rehti hai.

Reciprocal warning kyun. $\frac{dx}{dy}$ aur $\frac{dy}{dx}=g^{\prime }(x)$ ek doosre ke ulte hain: $\left|\frac{dx}{dy}\right|=\frac{1}{|g^{\prime }(x)|}.$ Formula ko inverse map ki slope chahiye, $\frac{dx}{dy}$. Reflex se $g^{\prime }(x)$ pakadna doosri classic mistake hai.

8. CDF $F_X$ — baayi taraf se count ki gayi paint

Picture. Ek jhaadu ko bahut baayi taraf se $x$ point tak sweep karo; $F_X(x)$ woh paint hai jo tumne sweep ki hai. Yeh $0$ se start hoti hai (kuch nahi sweep kiya) aur $1$ tak badhti hai (sab sweep kiya).

Topic ko iski zaroorat kyun hai. Densities ko directly manipulate karna fisalndaar ho sakta hai, lekin CDF hamesha theek behave karta hai. Parent ka sabse clean derivation kuch aisa hai: $F_Y(y)=P(Y\le y)=P(g(X)\le y)$, event ko $x$-statement mein translate karta hai, phir density wapas paane ke liye differentiate karta hai — kyunki density CDF ki slope hai: $f_X(x)=\frac{d}{dx}{F}_X(x).$

9. Total probability equals one — sanity check

Topic ko iski zaroorat kyun hai. Nayi $f_Y$ compute karne ke baad, usse integrate karne par phir bhi $1$ aana chahiye. Agar tumne Jacobian bhool diya, toh usually nahi aayega — toh yeh tumhara built-in error detector hai.

Foundations topic ko kaise feed karti hain

Ise aise padho: density ek slab of mass deti hai; transformation aur uska inverse batate hain slab kahan jaata hai; Jacobian batata hai kitna rescale hoga; us mass ka conservation hi technique hai.

Equipment checklist

Khud ko test karo — har line apna jawab reveal karti hai.