Joint distributions — joint PMF - PDF, marginal, conditional

4.9.10 · Maths › Probability Theory & Statistics

1. Joint distribution KYA hai?

Density kyun, probability kyun nahi? Continuous case mein kisi bhi single point ke liye $P (X = x, Y = y) = 0$ hota hai (ek point ka area zero hota hai). Toh hum probability per unit area describe karte hain; real probability tabhi milti hai jab kisi region $A$ par integrate karo.

Figure — Joint distributions — joint PMF - PDF, marginal, conditional

2. Marginals — ek variable ko nikaalna

"Marginal" naam isliye aaya kyunki joint PMF ko ek table ki tarah likhte hain aur row/column sums ko margins mein likhte hain.

3. Conditional — freeze karo aur re-normalize karo

4. Worked examples

Worked example Discrete joint table

Coin se related do variables ka joint PMF:

$p_{X, Y}$	$Y = 0$	$Y = 1$	row sum $p_{X}$
$X = 0$	$0.1$	$0.2$	$0.3$
$X = 1$	$0.3$	$0.4$	$0.7$
col sum $p_{Y}$	$0.4$	$0.6$	$1.0$

(a) $X$ ka Marginal: $p_{X} (0) = 0.1 + 0.2 = 0.3$ . Kyun? Row sum karo — $Y$ ko sum out karo. (b) Conditional $P (X = 1 ∣ Y = 0)$ : $\frac{p _{X, Y} ( 1 , 0 )}{p _{Y} ( 0 )} = \frac{0.3}{0.4} = 0.75$ . Kyun? Sirf $Y = 0$ column ( $0.1, 0.3$ ) dekho aur uske total $0.4$ se re-normalize karo. (c) Independent hain? Check karo $p_{X, Y} (0, 0) = 0.1$ vs $p_{X} (0) p_{Y} (0) = 0.3 \times 0.4 = 0.12$ . Barabar nahi $\Rightarrow$ dependent.

5. Common mistakes (steel-manned)

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek badi bucket ki grid hai. Har bucket mein kuch marbles hain ek pair (height, weight) ke liye. Joint yeh hai ki har bucket mein kitne marbles hain. Agar aap har row ke saare marbles ek saath push karo aur totals side mein likho, toh woh side-total height ka marginal hai. Ab maano main bolta hoon "weight exactly 50 kg hai" — aap ek column ko chhod kar baaki saari buckets phenk dete ho. Lekin bache hue marbles phir bhi ek poori bag ke barabar add up hone chahiye, toh aap unhe scale up karte ho. Yahi scaled-up column conditional hai.

Flashcards

Ek valid joint PMF ke liye kaun si do conditions honi chahiye?

p_{X, Y} (x, y) \geq 0

for all

x, y

, aur

\sum_{x} \sum_{y} p_{X, Y} (x, y) = 1

Joint se marginal PMF

p_{X}

kaise nikalte hain?

Doosre variable ko sum out karo:

p_{X} (x) = \sum_{y} p_{X, Y} (x, y)

Joint PDF se marginal PDF

f_{X}

kaise nikalte hain?

Doosre variable ko integrate out karo:

f_{X} (x) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d y

Conditional PDF

f_{X ∣ Y} (x ∣ y)

define karo.

f_{X ∣ Y} (x ∣ y) = f_{X, Y} (x, y) / f_{Y} (y)

for

f_{Y} (y) > 0

Conditional paane ke liye marginal se kyun divide karte hain?

Raw slice marginal tak integrate/sum hoti hai, 1 tak nahi; divide karna use valid distribution mein re-normalize karta hai.

X, Y

ke independent hone ki condition batao.

f_{X, Y} (x, y) = f_{X} (x) f_{Y} (y)

for all

x, y

(equivalently

f_{X ∣ Y} = f_{X}

Continuous variables ke liye

P (X = x, Y = y) = 0

kyun hota hai?

Ek single point ka area zero hota hai; sirf positive area wale regions probability carry karte hain density ke under.

Joint PDFs ka chain rule batao.

f_{X, Y} (x, y) = f_{X ∣ Y} (x ∣ y) f_{Y} (y) = f_{Y ∣ X} (y ∣ x) f_{X} (x)

f_{X, Y} = 3 x

ke liye

0 < y < x < 1

par,

f_{X} (x)

kya hai?

f_{X} (x) = \int_{0}^{x} 3 x d y = 3 x^{2}

0 < x < 1

Equal marginals — kya yeh independence imply karta hai?

Nahi; independence joint ke factoring ke baare mein hai, jo akele marginals se determine nahi ho sakti.

Connections

Conditional Probability — $P (A ∣ B) = P (A \cap B) / P (B)$ woh parent rule hai jahan se conditionals aate hain.
Independence of Random Variables — joint ke factorization se define hota hai.
Law of Total Probability — marginalization iska random-variable form hai.
Bayes' Theorem — $f_{X, Y} = f_{X ∣ Y} f_{Y} = f_{Y ∣ X} f_{X}$ se banta hai.
Covariance and Correlation — measure karta hai ki jointly distributed variables kaise co-vary karte hain.
Expectation and Variance — marginals/conditionals se compute hoti hain.
Bivariate Normal Distribution — ek key continuous joint example.