Covariance and correlation: Question bank — Hinglish deep dive

4.9.12 · D5 · Maths › Probability Theory & Statistics › Covariance and correlation

Neeche do shorthands use ho rahe hain (dono parent note mein define hain):

$Cov (X, Y) = E [(X - μ_{X}) (Y - μ_{Y})] = E [X Y] - E [X] E [Y]$ — mean se deviations ka average product.
$ρ_{X Y} = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}$ — wahi quantity jise dono standard deviations $σ_{X}, σ_{Y}$ se divide karke == $[- 1, 1]$ ki leash par rakha gaya hai==.

True or false — justify

Agar

ρ_{X Y} = 0

ho toh

X

aur

Y

independent hain.

False.

ρ = 0

sirf linear co-movement ko khatam karta hai; ek symmetric quadratic link jaise

Y = X^{2}

(jahan

X

0 ke baare mein symmetric ho) ka

ρ = 0

hoga phir bhi

Y

poori tarah

X

se determine hota hai. Independence of Random Variables dekho.

Agar

X, Y

independent hain toh

Cov (X, Y) = 0

True. Independence se

E [X Y] = E [X] E [Y]

milta hai, isliye

E [X Y] - E [X] E [Y] = 0

. Arrow sirf is direction mein chalta hai, kabhi wapas nahi.

500

ki covariance ka matlab

2

ki covariance se zyada strong relationship hai.

False. Covariance units carry karta hai aur variables ki magnitudes ke saath scale karta hai;

500 kg\cdotcm

aur

2 m\cdots

comparable nahi hain. Sirf unitless

ρ

hi strength measure karta hai.

Cov (X, X)

negative ho sakta hai.

False.

Cov (X, X) = E [X^{2}] - (E [X])^{2} = Var (X) \geq 0

; ek variable hamesha khud ke saath perfectly "agree" karta hai.

ρ_{X Y} = 1

ka matlab hai ki data points bilkul ek straight line par hain.

True.

ρ = \pm 1

Cauchy–Schwarz ka equality case hai, jo

Y = a X + b

exactly force karta hai;

+ 1

ka matlab positive slope,

- 1

ka matlab negative. Cauchy–Schwarz Inequality dekho.

X

- 3

se multiply karne par correlation ka sign flip ho jaata hai lekin magnitude nahi.

True.

ρ_{a X, Y} = sign (a) ρ_{X Y}

; factor

3

numerator aur denominator mein cancel ho jaata hai, sirf

a = - 3

ka sign bachta hai.

Var (X + Y) = Var (X) + Var (Y)

hamesha.

False. Ye

Var (X) + Var (Y) + 2 Cov (X, Y)

ke barabar hai; identity sirf tab hold hoti hai jab

Cov (X, Y) = 0

ho (jaise independence ke under).

Ice-cream sales aur drownings ke beech

0.9

correlation prove karta hai ki ice cream drowning ka cause hai.

False. Correlation causation nahi hai; ek lurking common cause (garmi ka mausam) dono variables ko bina kisi direct link ke saath upar drive kar sakta hai.

Har

X

ki value mein ek constant

b

add karne se

Cov (X, Y)

badal jaata hai.

False. Covariance location ko ignore karta hai:

Cov (X + b, Y) = Cov (X, Y)

kyunki shift

b

deviation

X - μ_{X}

ke andar cancel ho jaata hai.

ρ_{X Y} = 0.6

ka matlab hai ki

60%

points trend show karte hain.

False.

ρ

points ka percentage nahi hai; ye deviation products ka ek scaled average hai. "Variance explained" quantity

ρ^{2} = 0.36

hoti hai, aur woh bhi points ka count nahi hai. Linear Regression dekho.

Spot the error

Cov (2 X + 5, 3 Y - 1) = 2 \cdot 3 \cdot Cov (X, Y) + 5 - 1

Galat. Shifts kuch bhi contribute nahi karte: sahi value

2 \cdot 3 \cdot Cov (X, Y) = 6 Cov (X, Y)

hai. Constants

+ 5, - 1

bilinearity se gayab ho jaate hain.

"Kyunki

ρ \in [- 1, 1]

hai, covariance bhi

[- 1, 1]

mein rehti hai."

Galat. Sirf

ρ

bounded hai (ye rescaled version hai). Covariance unbounded hai aur koi bhi real number ho sakti hai, positive ya negative, kisi bhi magnitude ki.

X, Y

Cov = 0

hai, isliye

Var (X - Y) = Var (X) - Var (Y)

Galat. Variance kabhi subtract nahi hoti:

Var (X - Y) = Var (X) + Var (Y) - 2 Cov (X, Y)

, aur

Cov = 0

ke saath ye

Var (X) + Var (Y)

hai — ek sum.

ρ = - 1

ρ = + 1

se weaker hai kyunki ye negative hai."

Galat. Strength

∣ ρ ∣

hai;

- 1

aur

+ 1

dono perfectly linear hain. Sign sirf direction batata hai (opposite vs same), strength nahi.

"Unhone

Cov (X, Y) = E [X Y] - E [X] - E [Y]

compute kiya."

Galat. Subtract hone wala term means ka product hai, unka difference nahi:

Cov = E [X Y] - E [X] E [Y]

"Is dataset ke liye

ρ^{2} = 1.2

hai."

Galat. Impossible:

∣ ρ ∣ \leq 1

ρ^{2} \leq 1

force hota hai.

1

se upar ki value ek arithmetic error signal karti hai, shayad galat

σ

se divide kiya gaya ho.

Why questions

Hum dono deviations ko multiply kyun karte hain, jaise unhe add karne ki jagah?

Multiplication sabse sasta operation hai jo

+

return karta hai jab dono deviations ka sign same ho (agreement) aur

-

jab alag ho (disagreement); add karna sirf separate means recover karega aur co-movement kho dega.

Correlation paane ke liye covariance ko

σ_{X} σ_{Y}

se kyun divide karte hain?

Units aur scale strip out karne ke liye, taaki kg·cm aur rupees·seconds mein relationships comparable ho sakein; division Cauchy–Schwarz se result ko

[- 1, 1]

mein cap karta hai.

Cauchy–Schwarz proof

E [(U - t V)^{2}] \geq 0

se kyun shuru hoti hai?

Ek squared quantity kisi bhi

t

ke liye kabhi negative nahi hoti, isliye ye quadratic-in-

t

kabhi zero se neeche nahi ja sakta, jo uska discriminant

\leq 0

force karta hai — exactly inequality

Cov^{2} \leq σ_{X}^{2} σ_{Y}^{2}

. Cauchy–Schwarz Inequality dekho.

Covariance ko variance ka generalisation kyun kaha jaata hai?

Y = X

set karne par definition

E [(X - μ_{X})^{2}] = Var (X)

tak collapse ho jaati hai; variance sirf ek variable ki khud ke saath covariance hai. Variance and Standard Deviation dekho.

Independence

Cov = 0

force kyun kar sakti hai lekin

Cov = 0

independence force kyun nahi kar sakta?

Independence poori joint distribution control karti hai (

E [X Y] = E [X] E [Y]

aur bahut kuch), jabki covariance sirf linear/first-moment co-movement inspect karti hai — ek narrow slice jo zero ho sakta hai jabki nonlinear dependence bacha rahe.

Correlation swings kitni badi hain ye kyun ignore karta hai?

Kyunki har

σ

se divide karna har variable ko uski apni spread tak normalize kar deta hai, sirf how well they match chodta hai, how far they move nahi.

Do alag relationships compare karte waqt

Cov

ke upar

ρ

preferred kyun hai?

Cov

badal jaata hai agar aap sirf units rescale karo (

X

ko 1000 se multiply karo aur ye 1000× jump karta hai), jabki

ρ

scale-invariant hai, isliye ye genuine strength isolate karta hai.

Edge cases

ρ

kya hoga jab

Y

ek constant ho (maano

Y = 7

hamesha)?

Undefined. Ek constant ka

σ_{Y} = 0

hota hai, isliye hum zero se divide kar rahe honge; saath hi

Cov (X, 7) = 0

hai kyunki ek constant kabhi deviate nahi karta. Measure karne ke liye koi linear trend hi nahi hai.

Cov (X, Y)

kya hoga agar poori distribution ek single point

(a, b)

par ho?

Zero (actually degenerate): dono deviations hamesha

0

hoti hain, aur dono variances

0

hain, isliye covariance

0

hai aur

ρ

undefined hai.

Agar

Y = - 2 X + 7

exactly ho, toh

Cov (X, Y)

ka sign aur

ρ

kya honge?

ρ = - 1

(perfect negative linear);

Cov (X, Y) = - 2 Var (X) < 0

, negative kyunki slope

- 2

negative hai aur

Var (X) > 0

Kya

ρ = + 1

ho sakta hai jabki dono variables bilkul alag scales par hon (heights mm vs km mein)?

Haan.

ρ

scale-invariant hai, isliye ek perfect line

Y = a X + b

ρ = + 1

milega units ki parwah kiye bina — sirf slope

a

ka sign matter karta hai.

Symmetric

X

ke liye (maano uniform on

{- 1, 0, 1}

) aur

Y = ∣ X ∣

, kya

Cov = 0

hai?

Haan,

Cov = 0

, phir bhi

Y

poori tarah

X

se determine hota hai.

Y = X^{2}

ki tarah, dependence symmetric aur nonlinear hai, isliye ye covariance ko invisible hai.

ρ = - 1

ke extreme case mein

σ_{X} = σ_{Y} = σ

ke saath

Var (X + Y)

kya banega?

Var (X + Y) = σ^{2} + σ^{2} + 2 (- σ^{2}) = 0

— dono variables perfectly cancel kar dete hain, isliye unka sum ek constant hai jisme koi spread nahi.

Agar aap

X

ko factor

c \to \infty

se scale karo, toh

Cov (X, Y)

aur

ρ_{X Y}

ka kya hoga?

Cov

c Cov (X, Y) \to \pm \infty

ki tarah blow up karta hai, jabki

ρ

fixed rehta hai —

c

numerator aur

σ_{c X} = ∣ c ∣ σ_{X}

ke beech cancel ho jaata hai.

Connections

Covariance and Correlation — parent note jiske liye ye traps drill hain.
Variance and Standard Deviation — degenerate $σ = 0$ cases yahan hain.
Independence of Random Variables — in aadhe traps ke peeche ka one-way arrow.
Cauchy–Schwarz Inequality — kyun $∣ ρ ∣ \leq 1$ aur kyun $1.2$ impossible hai.
Linear Regression — $ρ^{2}$ variance explained ke roop mein, point count nahi.
Covariance Matrix — jahan ye pairwise facts vectors tak scale up hote hain.