Discrete random variables — PMF, CDF — Hinglish notes

4.9.3 · Maths › Probability Theory & Statistics

Discrete random variable KYA hota hai?

HUM kyun bother karte hain? Raw sample space $Ω$ (jaise "dice mein 3 aur 4 aaya") awkward hota hai. Hum usually ussi se nikala hua koi number chahte hain (jaise "sum 7 hai"). RV $X$ ek translator ka kaam karta hai — messy outcomes ko woh numbers mein badalta hai jinpar hum algebra kar sakein.

"Discrete" hone ka kya fayda hai? Kyunki values countable hain, hum probability ko individual points par assign kar sakte hain aur bas unhe add kar sakte hain. (Continuous RVs mein har point ki probability 0 hoti hai — wahan instead integrate karte hain. Yahi contrast is subtopic ke hone ki poori wajah hai.)

PMF — Probability Mass Function

Ek valid PMF ka behavior kaisa hona chahiye? Ise probability ke axioms se derive karte hain:

Probabilities kabhi negative nahi hoti, isliye $p_{X} (x) \geq 0$ for all $x$ . (Kyun? Axiom: $P (A) \geq 0$ .)
Events ${X = x_{i}}$ disjoint hain (X ek saath 2 aur 5 dono nahi ho sakta) aur milke woh sab kuch cover karte hain jo X kar sakta hai. Countable additivity se: $\sum_{i} p_{X} (x_{i}) = P (⋃_{i} {X = x_{i}}) = P (Ω) = 1.$

CDF — Cumulative Distribution Function

PMF se yeh kaise banta hai? Tum bas $\leq x$ wale saare points ka mass sum karte ho. Ek discrete RV ke liye yeh $F_{X}$ ko ek step function banata hai: values ke beech flat, aur har value $x_{i}$ par exactly $p_{X} (x_{i})$ ke barabar upar jump karta hai.

Figure — Discrete random variables — PMF, CDF

Interval probabilities — asli faayda

Worked examples

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek row mein buckets hain — 1, 2, 3… — aur tum total 1 liter paani dalte ho, game ke hisaab se buckets mein baant ke. PMF hai "har bucket mein kitna paani hai." Total hamesha exactly 1 liter hota hai (yahi rule hai). CDF hai "agar main bucket 1 se aage chalte hue paani add karta rehoon, toh is bucket tak pahunchte-pahunchte kitna collect ho gaya?" Jab koi bhara hua bucket milta hai toh total jump karta hai upar; buckets ke beech flat rehta hai. Aakhri bucket par hamesha poora 1 liter complete ho jaata hai.

Active recall

Ek random variable discrete kab hota hai?

Jab uski possible values ka set countable ho (finite ya countably infinite), taaki har value apna positive probability mass carry kar sake.

PMF

p_{X} (x)

define karo.

p_{X} (x) = P (X = x)

, yaani exact value

x

par probability mass.

Ek valid PMF ke liye kaunsi do conditions honi chahiye?

p_{X} (x) \geq 0

for all

x

, aur

\sum_{x} p_{X} (x) = 1

PMF ke terms mein CDF

F_{X} (x)

define karo.

F_{X} (x) = P (X \leq x) = \sum_{x_{i} \leq x} p_{X} (x_{i})

Ek discrete CDF ki shape kaisi hoti hai?

Ek right-continuous step (staircase) function: values ke beech flat, aur har value par

p_{X} (x_{i})

ke barabar upar jump karta hai.

CDF se PMF kaise recover karte hain?

p_{X} (x_{i}) = F_{X} (x_{i}) - F_{X} (x_{i}^{-})

— yaani

x_{i}

par jump ki size.

P (a < X \leq b)

ka formula do.

F_{X} (b) - F_{X} (a)

(

a

exclude,

b

include).

Kisi bhi CDF ki limiting values kya hoti hain?

F_{X} (- \infty) = 0

aur

F_{X} (+ \infty) = 1

, aur yeh non-decreasing hota hai.

CDF non-decreasing kyun hoti hai?

x

badhane par cumulative sum mein sirf non-negative probability mass add hoti hai; kuch kabhi subtract nahi hota.

p_{X} (k) = c k

k = 1, 2, 3

ke liye

c

nikalo.

6 c = 1 \Rightarrow c = \frac{1}{6}

, normalization

\sum p_{X} = 1

se.

Connections

Probability Axioms — countable additivity hi $\sum p_{X} = 1$ banati hai.
Continuous random variables — PDF, CDF — same CDF idea, lekin mass density ban jaata hai (sum ki jagah integrate karte hain).
Expectation and Variance of Discrete RVs — seedha PMF par based hai: $E [X] = \sum x p_{X} (x)$ .
Binomial Distribution, Poisson Distribution — named PMFs jo tum constantly use karoge.
Conditional Probability — conditional PMFs $p_{X ∣ Y}$ yahi machinery reuse karti hain.