Continuous random variables — PDF, CDF, percentiles

4.9.7 · Maths › Probability Theory & Statistics

1. Density kyun, probability kyun nahi?

Isliye hum $f (x)$ ki height se kabhi probability nahi padhte. Height density hai; area probability hai.

2. PDF — pehle principles se definition

KYA HAI: Ek function $f (x)$ jo describe karta hai ki probability real line par kaise spread hai.

YEH 1 tak integrate kyun karna chahiye: koi na koi value zaroor aayegi, isliye total area = total probability = 1.

3. CDF — density ko accumulate karna

KYA HAI: Cumulative distribution function $F (x)$ = "at most $x$ hone ki probability."

ISKO DEFINE KYUN KARTE HAIN: yeh "area" ke sawaalon ko simple subtraction mein convert karta hai aur har RV (discrete ya continuous) ke liye defined hai.

Iski properties ka derivation (PDF axioms se):

$F (- \infty) = \int_{- \infty}^{- \infty} f = 0$ , aur $F (\infty) = \int_{- \infty}^{\infty} f = 1$ .
$F$ non-decreasing hai: zyada area add karne se (kyunki $f \geq 0$ ) total shrink nahi ho sakta.
$F$ continuous RV ke liye continuous hai (koi jumps nahi, kyunki koi point mass carry nahi karta).

Figure — Continuous random variables — PDF, CDF, percentiles

4. Percentiles aur median

KYA HAI: $p$ -th percentile $x_{p}$ woh value hai jiske neeche probability ka fraction $p$ hota hai.

$F$ yahan invertible kyun hai: ek continuous RV ke liye jahan $f > 0$ apne support par hai, $F$ strictly increasing hai, isliye har $p$ ka ek unique $x_{p}$ hota hai.

7. Active-recall flashcards

f (x)

ko valid PDF banane ke liye kaunsi do conditions chahiye?

f (x) \geq 0

har jagah, aur

\int_{- \infty}^{\infty} f (x) d x = 1

Ek continuous RV ke liye

P (X = c)

kya hota hai?

0

, kyunki

\int_{c}^{c} f = 0

PDF se CDF kaise nikaalte hain?

F (x) = \int_{- \infty}^{x} f (t) d t

(left se area accumulate karo).

CDF se PDF kaise nikaalte hain?

f (x) = F^{'} (x)

, Fundamental Theorem of Calculus se.

P (a \leq X \leq b)

ko CDF se express karo.

F (b) - F (a)

p

-th percentile

x_{p}

define karo.

Woh value jahan

F (x_{p}) = p

; yaani

x_{p} = F^{- 1} (p)

Kya ek PDF value 1 se zyada ho sakti hai?

Haan — yeh ek density hai, sirf total area 1 ke barabar hona chahiye.

Continuous RVs ke liye

<

\leq

ka distinction irrelevant kyun hai?

Endpoints ka probability zero hota hai, isliye unhe include ya exclude karne se kuch nahi badalta.

f (x) = 2 x

[0, 1]

ka CDF?

F (x) = x^{2}

[0, 1]

λ

wale exponential ka median?

1 - e^{- λm} = 0.5

solve karo

\Rightarrow m = ln 2/ λ

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho 1 kg jam ek lambe bread ke piece par spread kiya gaya hai. Tum yeh nahi pooch sakte ki "exactly is ek dot par kitna jam hai?" — ek dot bahut chhotaa hota hai, jawab zero hoga. Lekin tum pooch sakte ho "yahan se wahan ke beech kitna jam hai?" — woh strip uthao aur tolo. PDF hai ki jam har jagah kitna mota spread hua hai. CDF hai ki "left edge se ab tak kitna jam mujhe mila hai." Median woh spot hai jahan exactly aadha jam ek taraf aur aadha doosri taraf ho.

Connections

Discrete random variables — PMF (discrete cousin: integrals ki jagah sums)
Expectation and Variance of continuous RVs ( $E [X] = \int x f (x) d x$ )
Normal distribution (iska CDF $Φ$ z-scores aur percentiles define karta hai)
Exponential distribution (worked Example 2; memorylessness)
Fundamental Theorem of Calculus ( $f \leftrightarrow F$ ka link)
Uniform distribution (constant density, linear CDF)
Quantile function and inverse-transform sampling ( $F^{- 1}$ use karke random numbers generate karna)

Continuous random variables — PDF, CDF, percentiles

1. Density kyun, probability kyun nahi?

2. PDF — pehle principles se definition

3. CDF — density ko accumulate karna

4. Percentiles aur median

5. Worked examples

6. Common mistakes

7. Active-recall flashcards

Connections

Concept Map

Deep Dive