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

4.9.10 · D1 · Maths › Probability Theory & Statistics › Joint distributions — joint PMF - PDF, marginal, conditional

Parent note par trust karne se pehle, tumhe page par har mark mein fluent hona chahiye. Yeh child har symbol ko zero se build karta hai — pehle plain words, phir ek picture, phir kyun yeh topic uske bina kaam nahi kar sakta. Upar se neeche padho; har brick neeche wali par tikti hai.

1. Random variable — $X$, $Y$

Ise picture karo. Ek die roll karo: woh machine jo ek number output karegi woh $X$ hai; woh number $3$ jo usne actually dikhaya woh $x$ hai. Capital slot hai; lower-case us slot mein ek value hai.

Topic ko iska kyun zaroorat hai. Poora chapter ek saath do aisi slots study karta hai — height $X$ aur weight $Y$ — isliye tumhe kabhi "variable" (capital) aur "uski ek value" (lower-case) mein confusion nahi honi chahiye. $P(X=x)$ jaisa har formula literally padhta hai "probability ki slot $X$ number $x$ par land kare."

2. Probability $P(\cdot )$ aur event $\{X=x\}$

Ise picture karo. Imagine karo experiment ke 100 repetitions 100 dots ki tarah hain. $P(X=x)$ un dots ka fraction hai jinke liye $X$ ka result $x$ aaya. $0$ = kabhi nahi, $1$ = hamesha, $0.5$ = aadhe dots.

Topic ko iska kyun zaroorat hai. Ek joint PMF inhi probabilities se bani hoti hai — $p_{X,Y}(x,y)=P(X=x\text{ and }Y=y)$. Agar "ek event ki probability" fuzzy hai, toh poori table fuzzy hai.

3. Do events ka "AND" — intersection

Ise picture karo (figure dekho). $x$ across aur $y$ upar wala plane draw karo. Event $\{X=x\}$ poori vertical line hai; $\{Y=y\}$ poori horizontal line hai. Unka AND woh single crossing point hai jahan dono satisfy hote hain.

Topic ko iska kyun zaroorat hai. Yeh crossing point joint ka ek cell hai. Joint distribution bas "har aisi crossing par kitni probability baithti hai" hai.

4. Disjoint events aur additivity

Ise picture karo. Do blobs jo overlap nahi karte. Kisi bhi blob mein land karne ka chance simply dono areas ka sum hai — koi double-counting nahi kyunki kuch bhi shared nahi hai.

Topic ko iska kyun zaroorat hai. Yahi marginals ke peeche ka engine hai. Events $\{X=x,Y=0\}$, $\{X=x,Y=1\}$, $\{X=x,Y=2\},\dots$ disjoint hain ($Y$ ki sirf ek value occur ho sakti hai), isliye unki probabilities add hoti hain $P(X=x)$ dene ke liye. Woh addition hi marginal formula hai. Yeh bilkul Law of Total Probability disguise mein hai.

5. Sum notation $\sum$

Ise picture karo. Ek table row: ${\sum }_y{p}_{X,Y}(x,y)$ = row ke saath slide karo, har cell add karte jao. Answer margin mein likha jaata hai — isliye "marginal."

Topic ko iska kyun zaroorat hai. Discrete marginals row/column sums hi hain: $p_X(x)={\sum }_y{p}_{X,Y}(x,y)$.

6. Integral $\int$ aur area-under-a-curve

Ise picture karo (figure dekho). Curve ke neeche shaded region; $dx$ ek skinny rectangle hai, aur $\int$ unhe sweep karke sum karta hai.

Topic ko iska kyun zaroorat hai. Continuous joint distributions mein, probability = density ke neeche volume/area, jo integrate karke milta hai. Marginals ban jaate hain $f_X(x)=\int f_{X,Y}(x,y)dy$: "row ke saath add karo" ka continuous version.

7. Density vs. probability — $f$ symbol

Ise picture karo (figure dekho). Plane ke upar baitha ek hill. Poore hill ka total volume $=1$. Ek patch ke upar volume $=$ us patch mein land karne ki probability.

8. Double integral $\iint$ aur support

Ise picture karo. Parent ke example ka triangle $0<y<x<1$ ek support hai: iske bahar hill ki height zero hai, isliye wahan integrals kuch contribute nahi karte. Limits of integration sahi karna = is footprint ka edge trace karna.

Topic ko iska kyun zaroorat hai. Wrong support ⇒ wrong limits ⇒ wrong marginals. Parent ki fourth "common mistake" exactly yahi bhoolna hai. Pehle hamesha footprint sketch karo.

9. Conditional bar $\mid$ aur re-normalization

Ise picture karo. Joint hill lo aur $Y=y$ par ek single slice kato. Us slice ka kuch total hota hai (uska area = $f_Y(y)$), lekin ek valid distribution ka total $1$ hona chahiye. Isliye tum slice ko scale up karte ho $f_Y(y)$ se divide karke. Yeh rescaled slice conditional hai. Yeh Conditional Probability par tikta hai: $P(A\mid B)=P(A\cap B)/P(B)$.

Topic ko iska kyun zaroorat hai. Conditioning slice + re-normalize hai, "Conditional $=$ Joint $÷$ Marginal" mein capture kiya gaya.

10. Marginal $f_X$, aur independence factoring

Topic ko inki kyun zaroorat hai. Marginals single-variable facts recover karti hain; independence woh special "no interaction" case hai jahan hill bas do 1D shapes ka outer product hai. Baad ke tools — Covariance and Correlation, Bayes' Theorem, Bivariate Normal Distribution, aur pairs par Expectation and Variance — sab directly inhi bricks par build karte hain.

Prerequisite map

Equipment checklist

Self-test: right side cover karo aur har ek answer karo. Agar koi trick kare, toh uska section dobara padho.