Foundations — Covariance and correlation

4.9.12 · D1 · Maths › Probability Theory & Statistics › Covariance and correlation

Ye page toolbox room hai. Covariance and Correlation mein jaane se pehle, hum har ek symbol, picture, aur idea yahan rakh dete hain jo wo note quietly assume karta hai. Upar se neeche padho — seedhi ke har paaydaan ka aadhaar uske neeche wala hai.

0. Kaccha maal: random variable $X$

Ek dartboard experiment socho. Har throw ek outcome hai; rule "measure karo dart centre se kitni door gira, cm mein" har throw ko ek number deta hai. Woh number-valued rule hi $X$ hai.

• Small letter $x$ ka matlab hai ek khaas value jo $X$ le sakta hai.
• Capital letter $X$ ka matlab hai poora random rule, result jaane se pehle.

Hum iska istemal isliye karte hain kyunki covariance do aise rules, $X$ aur $Y$, ko compare karta hai, jo usi experiment par measure kiye gaye hon (usi dart throw se horizontal position $X$ aur vertical position $Y$ dono milti hain).

Prerequisite depth Expectation of Random Variables aur Variance and Standard Deviation mein milti hai — hum yahan bas utna rebuild karenge jo zaroori hai.

1. Probability $P$: har value kitni likely hai

Average karne se pehle, hum chahiye wo symbol jo bata sake kitna likely hai.

Ek loaded die socho jiske faces $1,\dots ,6$ ki chances barabar nahi hain. Chheh heights ki list (ek har face ke liye), har ek $0$ aur $1$ ke beech aur sabka sum $1$, yahi pmf hai. Hum $P$ isliye chahiye kyunki neeche har average probability-weighted hai — probabilities nahi, averaging nahi.

2. Average: $E[\cdot ]$, aur mean $\mu$

Ek discrete variable ke liye jo value $x$ leti hai probability $P(X=x)$ ke saath: $E[X]={\sum }_{x}xP(X=x).$

Ise literally padho: har value $x$ par jaao jo variable le sakta hai, use uski likelihood se weight karo, add karo. $\sum$ (capital Greek "S", "sum" ke liye) ka matlab hai "saare cases par add karo".

Balance-point idea ko ek picture deserve hai.

Alag symbol kyun? Kyunki hum ise itni baar subtract karenge — $X-{\mu }_X$ — ki baar baar $E[X]$ likhne se page bhar jaata.

3. Deviation: $X-{\mu }_X$

Yahi sab kuch ka dil hai. Sign dekho:

• $X-{\mu }_X>0$ ⇒ $X$ above average nikla (ek accha din).
• $X-{\mu }_X<0$ ⇒ $X$ below average nikla (ek kharaab din).
• $X-{\mu }_X=0$ ⇒ $X$ exactly average par gira.

Mean kyun subtract karein? Covariance ko centre ke aas-paas movement se matlab hai, centre se nahi. Agar sab log ₹1000 zyaada kamate hain, kisi ka bhi "accha din / kharaab din" pattern nahi badlta — ${\mu }_X$ subtract karne se location chali jaati hai aur sirf wiggle bachti hai.

4. Ek deviation ko square karna: Variance $\mathrm{Var}(X)$ aur ${\sigma }_X$

Do variables compare karne se pehle, ek-variable wala case samjho jis par parent note build karta hai.

Square kyun? Ek raw deviation kabhi $+$, kabhi $-$ hoti hai; unhe average karne par zero ho jaata hai (yehi "balance point" ka matlab hai). Square karne se har deviation positive ho jaati hai, toh wiggles cancel nahi ho sakti. Badi wiggles count hoti hain; direction chali jaati hai.

Hum ${\sigma }_X$ aur ${\sigma }_Y$ isliye chahiye kyunki correlation covariance ko ${\sigma }_X{\sigma }_Y$ se divide karta hai units cancel karne ke liye. Zyaada detail Variance and Standard Deviation mein.

5. DO deviations ka multiplication: covariance aur uska shortcut

Ab ek variable se do mein chhalang. Do deviations $(X-{\mu }_X)(Y-{\mu }_Y)$ multiply karo aur product ka sign padho — yahi ek trick hai jis par poora topic tikaa hai.

Covariance phir bas is product ka average hai: $\mathrm{Cov}(X,Y)=E[(X-{\mu }_X)(Y-{\mu }_Y)].$ Agar agreements disagreements se zyaada hain, toh average positive hai. Yahi poori definition hai, ab puri tarah samjhi.

6. Correlation symbol $\rho$, uski $[-1,1]$ leash, aur yeh kahan toot ta hai

• $\rho =+1$: perfect agreement (points ek upar jaane wali line par girte hain).
• $\rho =-1$: perfect disagreement (points ek neeche jaane wali line par girte hain).
• $\rho =0$: koi linear teamwork nahi.

Leash kyun? Raw covariance kisi bhi size ki ho sakti hai aur units carry karti hai (kg·cm), toh "badi covariance" strength ke baare mein kuch nahi batati. Leash $\rho$ ko ek fair, unitless report card banaa deti hai.

7. Do supporting ideas jo topic inka sahara leta hai

Prerequisite map

Neeche ka diagram mermaid se draw kiya gaya hai (boxes aur arrows specify karne ka ek plain-text tarika). Har box ko ek concept aur har arrow ko "samajhna zaroori hai pehle" samjho: sabse neeche wale source boxes se shuru karo aur arrows ko upar follow karo — flow parent topic par khatam hota hai. Agar tumhara reader ise raw text dikhaye, toh sirf pairs ko words mein trace karo: random variable → expectation → mean → deviation, aur deviation se ladder do mein split hoti hai variance aur product of deviations, dono covariance ko feed karte hain, jo variance ke saath milkar correlation ko feed karta hai.

Seedhe words mein: probabilities hume average karne deti hain; averaging mean deta hai; mean hume deviations measure karne deti hai; deviations (squared) variance dete hain aur (paired, multiplied) covariance dete hain; covariance ko do sigmas se divide karne par correlation milta hai — doorway Covariance and Correlation mein. Sideways ideas Linear Regression aur Covariance Matrix ko bhi feed karte hain.

Equipment checklist

Apne aap ko test karo — right side cover karo.

Connections

• Expectation of Random Variables — woh $E[\cdot ]$ machine jo har formula use karta hai.
• Variance and Standard Deviation — ek-variable wala case jise covariance generalise karta hai.
• Independence of Random Variables — kyun independence $\mathrm{Cov}=0$ force karti hai.
• Cauchy–Schwarz Inequality — woh reason ki $\rho$ $[-1,1]$ mein rehta hai.
• Linear Regression — yahi deviations use karke ek line fit karta hai.
• Covariance Matrix — sabhi pairwise covariances ko ek saath bundle karta hai.
• Covariance and Correlation — parent topic jiske liye ye page prepare karta hai.