Common discrete distributions — Bernoulli, Binomial, Poisson, Geometric, Negative Binomial: Foundations

4.9.6 · D1 · Maths › Probability Theory & Statistics › Common discrete distributions — Bernoulli, Binomial, Poisson

Is page pe kuch bhi assume nahi kiya gaya. Har woh symbol jo parent note use karta hai, hum yahan introduce karte hain — ek-ek karke, is order mein ki har ek ke liye pichhla zaroori ho. Agar koi symbol parent page pe dikh raha hai aur tum 100% sure nahi ho ki uske chhote marks ka kya matlab hai — to woh neeche define kiya gaya hai.

0. Powers aur exponential — notation jo neeche har jagah use hoti hai

Kuch aur shuru karne se pehle, do notations hain jo parent page ki lagbhag har line pe aati hain: powers aur exponential function. Inhe hum abhi fix kar lete hain taaki baad mein kuch bhi surprise na ho.

Picture. $x^{k}$ baar baar stretching hai: $1$ se shuru karo aur $x$ factor se $k$ baar stretch karo. Hamare formulas mein $q^{k - 1}$ hai " $q$ ko khud se $k - 1$ baar multiply karo" — yani $k - 1$ independent failures ki chance ek ke baad ek.

Topic ko iske zaroori hone ki wajah. Independent probabilities ko multiply karne se $p^{k}$ aur $q^{n - k}$ jaise powers bante hain; Poisson limit se $e^{- λ}$ aata hai. Powers woh language hai jisme har distribution likhi jaati hai.

1. "Random variable" kya hota hai? Symbol $X$

Picture. Ek box socho (experiment). Andar kuch random hota hai — coin flip, die roll. Bahar ek number aata hai. $X$ is box par laga label hai; $k$ woh number hai jo is baar bahar aaya.

Figure — Common discrete distributions — Bernoulli, Binomial, Poisson, Geometric, Negative Binomial

Topic ko iske zaroori hone ki wajah. Har distribution is sawaal ka jawab deti hai: "kaun sa number aaya, aur kitna likely tha?" Agar "jo number aaya" ke liye koi symbol hi nahi hoga, toh sawaal bhi nahi poochh sakte. Dekho Bernoulli trial — woh sabse simple possible box hai.

2. Probability symbol $P (\cdot)$

Picture. Probability ko length 1 ki ek poori bar ki tarah socho jo saare possible outcomes mein baat jaati hai. Har slice ki width us outcome ki probability hai. Saari slices milke poori bar bharti hain.

Topic ko iske zaroori hone ki wajah. $P (X = k) = (k n) p^{k} q^{n - k}$ aur har dusra boxed formula ek slice ki width ka recipe hai. "Adds to 1" rule se hum un recipes ko sanity-check karte hain.

3. Success/failure numbers $p$ aur $q$

Picture. Length-1 bar ko exactly do pieces mein kaato: ek piece of width $p$ (outcome labelled $1$ ) aur bachi hui piece of width $q$ (outcome labelled $0$ ). Bas itna hi — yahi ek Bernoulli trial hai.

$q$ ko apna letter kyun milta hai. $q^{k - 1} p$ jaise formulas mein hum " $k - 1$ failures phir ek success" multiply karte hain. Har baar $(1 - p)$ likhne ki jagah $q$ likhna algebra ko readable rakhta hai — aur kuch nahi.

4. Counting: factorials $n!$ aur "choose" $(k n)$

Ab hume count karna hoga ki kitne arrangements same number of successes dete hain. Do symbols yeh kaam karte hain.

$n!$ ki picture. Tumhare paas $n$ empty slots aur $n$ distinct cards hain. Pehle slot mein $n$ choices hain, agले mein $n - 1$ , aur aage — multiply karte jao. Teen cards → $3! = 6$ orderings.

Picture. $n$ boxes ek line mein rakho. Exactly $k$ ko amber color karo (successes), baaki white rehne do. $(k n)$ amber boxes ke distinct patterns count karta hai.

Topic ko iske zaroori hone ki wajah. Binomial ka $(k n)$ count karta hai ki kaunse trials succeed hue; Negative Binomial ka $(r - 1 k - 1)$ count karta hai ki pehle ke kaunse trials succeed hue. Parent page pe har "choose" exactly yahi box-colouring picture hai.

5. Sum sign $\sum$ aur infinite sums

Picture. Ek conveyor belt terms $a_{1}, a_{2}, a_{3}, \dots$ ek running total mein daalti rehti hai. $\sum$ end mein bucket ka total hai.

Topic ko iske zaroori hone ki wajah. Geometric distribution ka mean $E [X] = \sum_{k = 1}^{\infty} k q^{k - 1} p$ ek infinite sum hai jo exactly is identity se collapse hoti hai (ek baar differentiate karke). Poori machinery ke liye dekho Geometric series.

6. Expectation $E [X]$ — "average value"

Picture. Number line pe $k$ position par $P (X = k)$ size ka weight rakho. $E [X]$ un saare weights ka balance point (centre of mass) hai.

Topic ko iske zaroori hone ki wajah. Har distribution apna mean report karti hai: Bernoulli $p$ , Binomial $n p$ , Geometric $1/ p$ . Yahi balance points hain. Poori treatment Expectation and Variance mein.

7. Variance $Var (X)$ — "spread"

Picture. Wapas number line pe weights ki taraf jaate hain. Variance measure karta hai ki weights balance point se kitne door hain, average pe (hum har distance ko square karte hain taaki left side ke weights, jo "negative distances" hain, right side wale ko cancel na karein).

Topic ko iske zaroori hone ki wajah. Har summary row mein variance list hoti hai. Parent note exactly is shortcut ko Bernoulli ke liye use karta hai (trick $X^{2} = X$ ke saath). Aur: variances tabhi add hoti hain jab trials independent hon — isliye parent note "independent trials" par insist karta hai. Details Expectation and Variance mein.

8. Constant $e$ aur limit $(1 + \frac{x}{n})^{n}$

Picture. £1 se shuru karo aur interest ek baar nahi balki $n$ ever-smaller instalments mein har saal add karo. Jaise $n \to \infty$ , total $e^{x}$ ke paas pahunchta hai — continuous compounding. Dekho Limit e^x as (1+x/n)^n.

Topic ko iske zaroori hone ki wajah. Poisson formula $P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}$ tab paida hota hai jab Binomial ka $(1 - \frac{λ}{n})^{n}$ $e^{- λ}$ ban jaata hai. $e$ nahi, Poisson nahi.

9. Rate $λ$

Picture. Ek time stretch (ya ek page, ya ek length) jisme chhote random dots bikre hain. $λ$ batata hai ki ek unit stretch mein kitne dots expect karte ho — yeh Poisson process ka viewpoint hai.

Topic ko iske zaroori hone ki wajah. $λ$ pair $(n, p)$ ko replace karta hai jab $n \to \infty$ : sirf product $n p = λ$ limit se survive karta hai.

Prerequisite map

Neeche ka diagram top-to-bottom padha jaata hai: random variable $X$ probability $P$ mein jaata hai, jo ek trial ko success $p$ aur failure $q$ mein split karta hai, aur yeh ek Bernoulli trial deta hai. Alag se, factorials choose symbol banate hain. Bernoulli aur choose milake Binomial dete hain; Bernoulli aur sum sign aur geometric series milake Geometric / Negative Binomial dete hain. Binomial ko uske limit tak push karne par $e$ milta hai, jo (rate $λ$ ke saath) Poisson deta hai. Finally expectation aur variance panon paanchon means aur variances ko feed karte hain.

Equipment checklist

Right side cover karo aur reveal karne se pehle zor se jawab do.

x^{k}

ka matlab kya hai kisi whole number

k

ke liye?

x

ko khud se

k

baar multiply karo;

x^{0} = 1

aur

x^{- k} = 1/ x^{k}

X

ka matlab kya hai, aur yeh

k

se kaise alag hai?

X

random machine hai (ek rule jo outcomes ko numbers mein convert karti hai);

k

ek specific number hai jo yeh output karta hai.

P (X = k)

ki saari values ko add karne par kya milna chahiye?

Exactly

1

— slices poori probability bar bhar deti hain.

Ek single trial ka

X

kaun se do numbers leta hai, aur unka kya matlab hai?

1

= success aur

0

= failure;

X \in {0, 1}

aur kuch nahi.

Agar

p

success chance hai, to

q

kya hai?

q = 1 - p

, failure chance; pure shorthand, koi nayi info nahi.

(k n)

kin

n

aur

k

ke liye defined hai, aur agar

k

range se bahar ho to?

Non-negative whole numbers jisme

0 \leq k \leq n

; us range se bahar

(k n) = 0

(k n)

kya count karta hai, picture mein?

n

boxes mein se

k

ko successes ki tarah colour karne ke distinct tareekon ki ginti, order ignore karke.

(k n)

ko factorials mein likho.

\frac{n !}{k ! ( n - k )!}

∣ x ∣

ka matlab kya hai, aur

∣ x ∣ < 1

kya kehta hai?

x

ki size sign ignore karke;

∣ x ∣ < 1

matlab

- 1 < x < 1

∣ x ∣ < 1

ke liye

\sum_{k = 0}^{\infty} x^{k}

kya hai?

\frac{1}{1 - x}

(geometric series).

E [X] = \sum_{k} k P (X = k)

mein

k

kin values ke upar run karta hai?

Har woh value jo

X

le sakta hai (uska poora support) — finite ya infinite.

Constant

c

ke liye

E [c]

kya hai, aur

E [c X]

E [c] = c

aur

E [c X] = c E [X]

— constants average se seedhe guzar jaate hain.

Shortcut

Var (X) = E [X^{2}] - (E [X])^{2}

kyun hold karta hai?

(X - μ)^{2}

expand karo aur average lo; constant

μ

middle terms ko

- μ^{2}

mein collapse kar deta hai.

E [X]

kis cheez ka balance point hai?

Number line pe position

k

par rakhe

P (X = k)

size ke weights ka.

Binomial mean instantly

n p

kyun likha ja sakta hai?

Linearity of expectation — averages bina independence ke bhi add hote hain, aur yeh

n

Bernoulli copies hain.

Variances add hone ke liye kaun si extra condition chahiye?

Trials ki independence.

n \to \infty

ka matlab kya hai, aur limit kya hoti hai?

n

bina ruke badhta hai; limit woh fixed number hai jis par ek sequence settle hoti hai.

n \to \infty

ke saath

(1 + \frac{x}{n})^{n}

kya approach karta hai?

e^{x}

Poisson limit mein

(n, p)

ko kaunsa ek number replace karta hai, aur kaise?

λ = n p

, jo fixed rakhaa jaata hai jaise

n \to \infty

p \to 0

Taiyar ho? parent note pe har symbol ke paas ab plain-words meaning aur ek picture hai.