Joint distributions — joint PMF - PDF, marginal, conditional: Visual walkthrough

4.9.10 · D2 · Maths › Probability Theory & Statistics › Joint distributions — joint PMF - PDF, marginal, conditional

Step 1 — Ek saath do numbers: ek "point" ka matlab

KYA. Hamare paas do random quantities hain. Pehli ko $X$ kaho aur doosri ko $Y$ . Ek single outcome sirf ek number nahi balki ek pair $(x, y)$ hota hai — flat board par ek dot. Horizontal position $X$ ki value hai; vertical position $Y$ ki value hai.

KYUN. Yeh poochhne se pehle ki " $X$ aur $Y$ saath saath kaise chalte hain," humein ek aisa stage chahiye jahaan dono ek saath rahein. Ek number ek line par rehta hai; do number ek plane par rehte hain. Woh plane hi poora khel hai.

PICTURE. Figure s01 dekho. Pale-yellow dot $x = 3$ (teen kadam daayein) aur $y = 2$ (do kadam upar) par baitha hai. Dot padhne ka matlab hai dono coordinates padhna.

Step 2 — Probability ki dheri lagana: the joint

KYA. Ab socho experiment laakhon baar run kiya aur har observe kiye gaye $(x, y)$ par ek marble gira diya. Kuch spots par unchi stacks ban gayi, kuch khaali rahi. $(x, y)$ ke upar stack ki unchai bataati hai woh pair kitna likely hai. Woh height function hi joint hai.

KYUN. Hum chahte hain ki ek object sab kuch store kare. Yeh nahi ki "kitna likely hai $X = 3$ " aur alag se "kitna likely hai $Y = 2$ ", balki "exactly pair $(3, 2)$ kitna likely hai." Stack ki height seedha yeh bataati hai.

PICTURE. Figure s02 mein board ek grid of buckets hai (discrete case). Har bucket mein jo number hai woh hai $p_{X, Y} (x, y)$ — marbles ka uska hissa. Do rules picture se seedhe dikhte hain:

Koi bhi bucket negative pile nahi rakh sakta → $p_{X, Y} (x, y) \geq 0$ .
Saare buckets milake poora bag rakhte hain → numbers ka sum $1$ hai.

height over the bucket (x, y) p_{X, Y} (x, y) = share of marbles landing there P (X = x and Y = y), no negative piles p_{X, Y} \geq 0, whole bag x \sum y \sum p_{X, Y} = 1 .

Step 3 — Dheri ko flat karna: the marginal

KYA. Maan lo mujhe ab $Y$ ki parwah nahi. Main sirf poochhta hoon: " $X = x$ kitna likely hai, $Y$ kuch bhi kare?" Iske liye row $X = x$ ka har marble ek side-pile mein dhakel deta hoon aur uska total dekhta hoon.

KYUN. " $X = x$ , $Y$ kuch bhi" woh event hai ${X = x, Y = y_{1}}$ ya ${X = x, Y = y_{2}}$ ya … Yeh sub-events kabhi overlap nahi karte (ek marble ka sirf ek $y$ hota hai), aur milke yeh poori row hain. Non-overlapping events ke union ki probability = unki probabilities ka sum. Yahi Law of Total Probability ka kaam hai.

PICTURE. Figure s03 mein arrows dikhate hain: row ke har bucket ko baayein slide karke margin mein daal diya — table ke edge par likhi strip. Woh edge total marginal hai. (Literally wahi se word marginal aaya hai: margin mein likha jata tha.)

margin total for row x p_{X} (x) = collapse all y y \sum each bucket in the row p_{X, Y} (x, y)

$\sum_{y}$ = "har $y$ par sweep karo aur jodo." Yeh collapse hai, deletion nahi.
Result sirf $x$ par depend karta hai — $Y$ chala gaya.

Step 4 — Ek slice ko freeze karna: raw slice kyun toot jaati hai

KYA. Ab ulta move. Koi mujhe bataata hai $Y = y_{0}$ exactly. Main column $Y = y_{0}$ ke alawa har bucket phenk deta hoon aur jo bacha hai usse dekhta hoon.

KYUN. $Y = y_{0}$ jaanna duniya ko chhota kar deta hai: sirf usi $y$ wale outcomes possible hain. Column hi nayi possibilities ka set hai.

PICTURE. Figure s04 mein poora board gray hai siwaaye ek bright chalk-blue column ke. Lekin column ke neeche likha number dekho — bache hue marbles $p_{Y} (y_{0})$ tak add hote hain, jo ki 1 se kam hai. Ek distribution ka total zaroor 1 hona chahiye. Toh yeh slice, jaise hai, abhi legal distribution nahi hai.

x \sum buckets left in the column p_{X, Y} (x, y_{0}) = column total < 1 p_{Y} (y_{0}) \Rightarrow not yet a distribution.

Step 5 — Slice ko re-inflate karna: the conditional

KYA. Har bache hue bucket ko same factor se scale karo taaki column phir se $1$ ho jaaye. Yeh kaam karne wala factor hai column total $p_{Y} (y_{0})$ se divide karna.

KYUN. "Given" ke matlab se shuru karo — Conditional Probability rule $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ . $A = {X = x}$ aur $B = {Y = y_{0}}$ rakho: intersection bucket $p_{X, Y} (x, y_{0})$ hai, aur $P (B)$ column total hai. Column total se divide karna exactly re-inflation hai, aur yeh buckets ke beech ke ratios intact rakhta hai.

PICTURE. Figure s05 tall re-scaled column ko short raw column ke paas rakhta hai. Same shape, lekin tall taaki ab probability ki ek poori unit bhar sake. Green check confirm karta hai ki sum $1$ hai.

re-inflated column, sums to 1 p_{X ∣ Y} (x ∣ y_{0}) = \frac{p _{X, Y} ( x , y _{0} ) one bucket in the column}{column total (the scale factor) p _{Y} ( y _{0} )}, p_{Y} (y_{0}) > 0.

Step 6 — Ulta padhna: the chain rule

KYA. Step 5 mein column total se multiply karo. Division undo ho jaata hai aur product ban jaata hai.

KYUN. Kabhi kabhi tum joint ko ek story se build karte ho: "pehle $Y$ draw karo, phir $Y$ given $X$ draw karo." Woh story ek multiplication hai, aur yeh sirf Step 5 ka rearrangement hai. Dono orders same joint deni chahiye.

PICTURE. Figure s06 mein do-arrow factory dikhayi hai: ek column chuno weight $p_{Y} (y)$ ke saath, phir uske andar ek bucket chuno re-inflated height $p_{X ∣ Y} (x ∣ y)$ ke saath; unka product original bucket rebuild karta hai.

original bucket f_{X, Y} (x, y) = column ke andar shape f_{X ∣ Y} (x ∣ y) \cdot column kitna bhaari hai f_{Y} (y) = f_{Y ∣ X} (y ∣ x) f_{X} (x) .

Step 7 — Degenerate case: independence

KYA. Special situation: $Y$ jaanna $X$ ke baare mein kuch nahi bataata. Tab re-inflated column har $y$ ke liye identical dikhti hai — $Y$ freeze karne se shape kabhi nahi badli.

KYUN. " $Y$ seekhna kuch nahi badalta" ka matlab hai $f_{X ∣ Y} (x ∣ y) = f_{X} (x)$ . Ise Step 6 ke chain rule mein daalo: $f_{X, Y} = f_{X} \cdot f_{Y}$ . Joint apne dono marginals mein factorize ho jaati hai — dekho Independence of Random Variables.

PICTURE. Figure s07 do boards contrast karta hai. Left: har column ka same profile hai → surface ek clean outer product hai, independent. Right: profile $y$ ke saath tilt hoti hai → dependent. Tum literally dependence ko column-to-column shape change ke roop mein dekhte ho.

Independent ⟺ joint ek product mein split hoti hai f_{X, Y} (x, y) = f_{X} (x) f_{Y} (y) \forall x, y ⟺ column shape kabhi nahi badlati f_{X ∣ Y} (x ∣ y) = f_{X} (x) .

Step 8 — Live dekho: continuous triangle example

KYA. Poora pipeline ek baar real dust par chalao: $f_{X, Y} (x, y) = 3 x$ triangle $0 < y < x < 1$ par.

KYUN. Har rule upar wala, order mein, ek aisi case par jahan support matter karta hai — sloped edge $y < x$ integration limits decide karti hai.

PICTURE. Figure s08 triangle draw karta hai. Horizontal blue arrow ( $X$ ke marginal ke liye) $y$ ko $0$ se sloped edge $y = x$ tak sweep karta hai. Vertical pink arrow ( $Y$ ke marginal ke liye) $x$ ko edge $x = y$ se $1$ tak sweep karta hai. Limits picture se padhte hain.

Marginal of $X$ ( $y$ collapse karo, $0$ se $x$ tak): $f_{X} (x) = \int_{0}^{x} 3 x d y = 3 x \cdot x = 3 x^{2}, 0 < x < 1.$
Marginal of $Y$ ( $x$ collapse karo, $y$ se $1$ tak): $f_{Y} (y) = \int_{y}^{1} 3 x d x = \frac{3}{2} (1 - y^{2}), 0 < y < 1.$
Conditional (divide karo, support $y < x < 1$ ho jaata hai): $f_{X ∣ Y} (x ∣ y) = \frac{3 x}{\frac{3}{2} ( 1 - y ^{2} )} = \frac{2 x}{1 - y ^{2}} .$

Ek-picture summary

Figure s09 poori journey ek board par rakhta hai: centre mein joint surface, har axis par flattened marginals (sum/integral arrows), ek frozen column utha ke conditional mein re-inflate hoti hui, aur product-arrows chain rule dikhate hue. Yeh single diagram hi parent note hai.

Recall Feynman retelling — plain words mein poora walkthrough

Ek board of buckets socho, har ek mein kuch marbles hain ek pair $(x, y)$ ke liye. Woh poori dheri joint hai — ise sab kuch pata hai. Ek row ke har bucket ko saath dhakel kar margin mein total likho: woh $X$ ka marginal hai — tumne $Y$ ignore nahi kiya, tumne uske sab pe add kiya. Ab koi kehta hai " $Y$ exactly $y_{0}$ hai." $Y = y_{0}$ column ke alawa har bucket phenk do. Lekin bachey hue marbles poora bag nahi bharte — sirf column ke apne weight tak add hote hain. Toh sabko us weight se divide karke scale up karo; ab woh ek proper bag bharte hain. Woh re-inflated column conditional hai. Last step ulta chalao — column ka weight times uske andar ki shape — aur koi bhi bucket rebuild karo: woh chain rule hai. Aur agar har column ki shape exactly same hai chahe tum kaunsa bhi $y$ freeze karo, tab $Y$ ne $X$ ke baare mein kuch nahi bataya: dheri apne dono margins ke plain product mein split ho jaati hai — woh independence hai. Teen moves, ek dheri: jar, margins, slice-and-rescale.

Connections

Joint distributions — joint PMF - PDF, marginal, conditional — woh parent jise yeh page derive karta hai.
Conditional Probability — woh $P (A \cap B) / P (B)$ rule jo Step 5 ko power karta hai.
Law of Total Probability — Step 3 mein union-of-disjoint-events sum.
Independence of Random Variables — Step 7 ka factorization.
Bayes' Theorem — Step 6 ke chain rule ko reverse karna.
Covariance and Correlation — dependent surfaces (Step 7 right) ko quantify karna.
Expectation and Variance — in distributions par average lena next.
Bivariate Normal Distribution — woh smooth joint jahaan har conditional phir se normal hoti hai.