Foundations — Continuous random variables — PDF, CDF, percentiles

4.9.7 · D1 · Maths › Probability Theory & Statistics › Continuous random variables — PDF, CDF, percentiles

Yeh page assume karta hai ki aapne parent note ki koi bhi notation nahi dekhi. Hum har symbol ko scratch se build karte hain, us order mein jisme woh ek doosre par depend karte hain, taaki jab aap ${\int }_{-\infty }^{x}f(t)dt$ se milein toh aapke paas uska har piece pehle se ho.

0. "Random variable" aakhir hai kya?

Capital-vs-lowercase ka difference ek aisi aadat hai jo rakhni chahiye: $X$ machine hai, $x$ ek output hai jis par aap point karte ho. Jab hum $P(X\le x)$ likhte hain toh matlab hai "machine ke output ke us particular number $x$ par ya usse neeche aane ka chance."

Do flavours exist karte hain:

• Discrete — alag-alag dots par land karta hai ($X=1,2,3,\dots$). Dekho Discrete random variables — PMF.
• Continuous — ek range mein kahin bhi land kar sakta hai (kisi insaan ki exact height, $172.4839\dots$ cm). Yeh poora topic continuous flavour ke baare mein hai.

1. Symbol $P(\cdot )$ — "yeh probability hai ki…"

Brackets ke andar ek $X$ ke baare mein statement aati hai:

• $P(X\le 3)$ = output ke zyada se zyada $3$ hone ka chance.
• $P(2\le X\le 5)$ = output ke $2$ aur $5$ ke beech land karne ka chance.

Symbol $\le$ ka matlab hai "less than or equal to"; $<$ ka matlab hai "strictly less than." Yeh distinction yaad rakho — continuous variables ke liye yeh matter karna band ho jaayega, aur hum D2 mein prove karenge kyun, lekin pehle aapko dono symbols alag se pata hone chahiye.

2. Ek curve ke neeche Area, aur kyun humein $\int$ chahiye

Yahan discrete se continuous ka leap hai. Dots ke liye, aap har dot ke chances add karte ho. Ek continuous range ke liye infinitely many values ek saath pack hoti hain, toh ek-ek karke add karna hopeless hai. Iske bajaaye hum area measure karte hain.

Yeh tool kyun aur plain sum $\sum$ kyun nahi? Ek sum $\sum$ ek countable list of separate items add karta hai — dots ke liye perfect. Lekin ek continuous range mein koi "next" value nahi hoti jis par step karo; kisi bhi do numbers ke beech ek aur number hoti hai. Integral woh ek tool hai jo ek smooth, gap-free quantity ko total karne ke liye bana hai. Yahi reason hai ki continuous probability integrals par jeeti hai aur discrete probability sums par.

3. Density $f(x)$ — curve ki height

Critical warning, ab batate hain taaki yeh kabhi surprise na kare:

Mass-density analogy ise concrete banati hai:

4. Woh do rules jo $f$ ko ek probability density banate hain

Kyunki area probability hai aur total probability $1$ hai, do rules turant nikalte hain:

Symbols $-\infty$ aur $+\infty$ ("negative infinity", "positive infinity") ka matlab hai "saare real numbers mein sweep karo, bahut left se bahut right tak." Hum inhe isliye use karte hain kyunki $X$ principle mein koi bhi real number ho sakta hai; curve simply $0$ hoti hai jahan $X$ land nahi kar sakta.

5. Symbol $F(x)$ — left se running total

Ab hum accumulate karte hain. Ek single strip ke baare mein poochne ki jagah, pucho: "bahut left se sweep karte hue point $x$ tak kitna area collect kiya?"

Do notational points jo beginners ko trip karte hain:

• Andar dummy letter $t$ kyun? Integral ki upper limit pehle se $x$ kahlati hai. Hum $x$ ko sweeping variable ke roop mein dobara use nahi kar sakte, warna symbol ke do matlab ho jaate. Toh hum ek throwaway naam $t$ se sweep karte hain; integrate karne ke baad woh disappear ho jaata hai. $F(x)={\int }_{-\infty }^{x}f(t)dt$ aur ${\int }_{-\infty }^{x}f(s)ds$ ek hi cheez hain.
• Capital $F$ kyun? Convention: lowercase $f$ = small local density; uppercase $F$ = big accumulated total. "Chhota letter, chhota idea (yahan thickness); bada letter, bada idea (ab tak sab kuch)."

Picture se aap $F$ ki har property read kar sakte ho:

• Bahut left par, abhi koi area collect nahi hua: $F(-\infty )=0$.
• Bahut right par, poora area collect ho gaya: $F(+\infty )=1$.
• Rightward sweep karna sirf area add karta hai (kyunki $f\ge 0$), toh $F$ kabhi neeche nahi jaata — yeh non-decreasing hai.

6. Link $f\leftrightarrow F$: ek taraf integrate karo, wapas differentiate karo

Ab tumhare paas same variable ke liye do curves hain: density $f$ aur uska running-total $F$. Yeh ek coin ke do sides hain, Fundamental Theorem of Calculus se connected.

Naya symbol $\frac{d}{dx}$ (ya $F^{\prime }$) hai derivative — yeh $F$ ke slope / rate of change ko measure karta hai.

7. Inverse $F^{-1}$ — "kitna?" ki jagah "kahan?" poochna

Ek percentile sawaal ko flip kar deta hai. Normally hum $F$ ko ek value $x$ dete hain aur woh ek probability $p$ return karta hai. Ek percentile ek probability $p$ de deta hai aur poochta hai: kaunse $x$ ne ise produce kiya?

$F^{-1}$ exist karne ki guarantee kyun hai? Sirf woh functions jo kabhi koi value repeat nahi karte, invert ho sakte hain. Kyunki $F$ strictly increasing hai jahan $f>0$ hai, har probability $p$ exactly ek $x$ se correspond karti hai — koi ambiguity nahi. Yahi cheez "90th percentile" ko ek single well-defined number banati hai. $F^{-1}$ ko random numbers generate karne ke liye use karna Quantile function and inverse-transform sampling mein hai.

8. Woh symbols jo worked examples mein milenge

Parent note ke examples kuch aur pieces lete hain. Inhe abhi grab karo taaki baad mein kuch naya na ho.

Aur everyday shorthand jo dikhegi: ​ (square root, squaring ko undo karta hai), $[g(x){]}_a^b=g(b)-g(a)$ (expression ko top limit par evaluate karo minus bottom limit — yeh har integral ka "plug in and subtract" step hai).

Prerequisite map

Equipment checklist

Khud test karo — har line ki right side cover karo.

Connections

• Discrete random variables — PMF (is poore page ka dots-and-sums cousin)
• Fundamental Theorem of Calculus ($f=F^{\prime }$ aur $F=\int f$ ke peechhe ki engine)
• Exponential distribution (jahan $e^{-\lambda x}$ aur $\ln$ aate hain)
• Uniform distribution (inhe test karne ke liye sabse simple constant-density example)
• Quantile function and inverse-transform sampling ($F^{-1}$ ko earnest mein use karna)
• Expectation and Variance of continuous RVs (isi $f$ par build hone wale next tools)
• Normal distribution (iski CDF $\Phi$ ek percentile machine hai)