Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta: Question bank

4.9.8 · D5 · Maths › Probability Theory & Statistics › Common continuous distributions — Uniform, Normal, Exponenti

Shuru karne se pehle, ek word jis par hum baar baar depend karte hain:

PDF = probability density function $f (x)$ : curve ki height, probability NAHI.
CDF = cumulative distribution function $F (x) = P (X \leq x)$ : left se running area.

Neeche chaar chhoti galleries hain jo un shapes ko anchor karti hain jinhe questions baar baar refer karte hain — pehle inhe dekh lo taaki "hump", "U-shape", "J-shape", aur "spike" words ka kuch concrete matlab ho.

Figure — Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta

True or false — justify

A "flat" PDF means every value is equally likely, so the probability of any single point is nonzero

False — ek continuous variable ke liye probability of any single exact point is zero; "equally likely" ka matlab equal density hai, aur probability tab milti hai jab tum ek interval pe integrate karo.

U (a, b)

ke liye, width

b - a

ko double karne par variance bhi double ho jaata hai

False — variance hai

(b - a)^{2} /12

, toh width double karne par variance

4

se multiply hota hai,

2

se nahi.

Kisi point par PDF

1

se bada ho sakta hai

True — sirf area

1

ke barabar hona chahiye; jaise

U (0, 0.5)

ki height har jagah

2

hai. Density probability nahi hai, isliye uski koi upper bound nahi.

Exponential distribution ki memorylessness ka matlab hai ki event "reset" ho jaata hai aur jitna zyada wait karo, utna zyada likely ho jaata hai

False — memoryless ka matlab hai future wait ki same distribution hogi chahe kitna bhi time beet gaya ho; na zyada likely hota hai na kam, bas bhool jaata hai.

Agar

X \sim Gamma (α, λ)

aur

α

integer hai, toh

X

α

independent

Exp (λ)

waits ka sum hai

True — Gamma

α

-ve event ka wait hai, jo

α

successive first-event waits ka sum hai, har ek

Exp (λ)

Kyunki Normal symmetric hai,

P (X \leq μ)

hamesha

0.5

hoti hai

True —

μ

ke aas paas symmetry total area

1

ko mean par do equal halves mein baant deti hai, har

μ

aur

σ

ke liye.

Beta (1, 1)

sirf

[0, 1]

par uniform distribution hai

True —

α = β = 1

plug karo: density ban jaati hai

x^{0} (1 - x)^{0} / B (1, 1) = 1

[0, 1]

par ek flat line.

Exp (λ)

mein bada rate

λ

matlab longer average waiting times

False — mean hai

1/ λ

, toh higher rate matlab events jaldi aate hain aur average wait is shorter.

Beta ka mean

α / (α + β)

agar

α

bada ho toh

1

se zyada ho sakta hai

False — kyunki

α, β > 0

hain, fraction

α / (α + β)

hamesha strictly

0

aur

1

ke beech hota hai, jo support

(0, 1)

se match karta hai.

Same variance lekin alag means wale do Normals ki curves ka shape alag hoga

False —

μ

badlane se curve sirf slides the curve horizontally; shape (width, height) sirf

σ

se set hota hai, jo unchanged hai.

Spot the error

Var (U (a, b)) = (b - a)^{2} /4

because the range is

b - a

and I halve it like a radius."

Sahi jawab hai

(b - a)^{2} / 12

;

12

aata hai

\frac{1}{3} - \frac{1}{4}

wali algebra se

E [X^{2}] - μ^{2}

mein, range half karne se nahi.

"For Normal I plug

σ^{2}

into the denominator, so the density is

\frac{1}{σ ^{2} 2 π} exp (\dots)

Galat — normalizer

σ 2 π

mein aur standardization

z = (x - μ) / σ

mein ==standard deviation

σ

== hota hai (na ki

σ^{2}

P (X \leq 185)

for

N (170, 1 0^{2})

Φ (185)

, I just read the table at 185."

Pehle standardize karna padega:

z = (185 - 170) /10 = 1.5

, phir

Φ (1.5)

dekho; tables sirf standard

N (0, 1)

ke liye hain.

"Exponential mean equals its parameter

λ

, since

λ

is the only knob."

Mean hai reciprocal

1/ λ

;

λ

ek rate hai (events per unit time), mean ek time hai.

"Gamma

(α, λ)

has mean

α λ

because both parameters push it up."

Mean hai

α / λ

; zyada required events (

α

) wait badhate hain, lekin higher rate (

λ

) use ghataata hai, isliye

λ

divide karta hai.

"Beta

(2, 5)

leans right (toward 1) because 5 is bigger."

Ye left lean karta hai (toward 0): bada

β

near 0 pe "failure mass" dher karta hai, mean

2/7 \approx 0.286

deta hai.

"Since

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

, the CDF's slope can be negative where the density dips."

CDF har jagah non-decreasing hoti hai, isliye uska slope

f (x) \geq 0

hota hai; density kabhi negative nahi hoti — ek "dip" matlab ek chhota positive slope hai, na ki downhill CDF.

Why questions

Why does the exponential density

λ e^{- λ t}

come from the Poisson "zero events" probability?

"Time

t

tak koi event nahi" matlab survival

P (X > t) = e^{- λ t}

hai (Poisson at

k = 0

);

F = 1 - e^{- λ t}

differentiate karne par density milti hai. Dekho Poisson Process.

Why is

2 π

the magic constant in the Normal density?

Ye Gaussian integral

\int_{- \infty}^{\infty} e^{- z^{2} /2} d z = 2 π

ki value hai, isliye isse divide karne par total area

1

ho jaata hai.

Why does the Central Limit Theorem make the Normal appear everywhere?

Bahut saare independent chhote effects ke sums/averages bell shape mein converge karte hain, chahe individual pieces ki distribution kuch bhi ho — dekho Central Limit Theorem.

Why must every PDF integrate to exactly

1

, not just be non-negative?

Sab outcomes ki total probability

1

honi chahiye;

f

ke neeche ka area wohi total probability hai, isliye ye

1

pe pin hai.

Why does the Beta function

B (α, β)

appear as the Beta density's denominator?

B (α, β) = \int_{0}^{1} x^{α - 1} (1 - x)^{β - 1} d x

exactly un-normalized shape ke neeche ka raw area hai, isliye isse divide karne par density

1

integrate karti hai. Dekho Beta Function.

Why does the Gamma function

Γ (α)

let us handle non-integer "number of events"?

Γ

factorial ko (

Γ (n) = (n - 1)!

) smoothly sab

α > 0

tak interpolate karta hai, isliye Gamma distribution tab bhi defined hai jab

α

poori count nahi hai. Dekho Gamma Function.

Why is exponential a special case of Gamma but not vice versa?

Gamma ki density mein

α = 1

set karne par

x^{α - 1}

collapse hokar

1

ban jaata hai aur

λ e^{- λ x}

milta hai; general Gamma with

α \neq = 1

mein aisa collapse nahi hota, isliye ye strictly zyada general hai.

Edge cases

U (a, b)

ka kya hoga jab

a \to b

(interval ek point tak shrink ho jaaye)?

Height

1/ (b - a) \to \infty

aur width

\to 0

, toh koi ordinary PDF survive nahi karta; limit hai ek Dirac delta

δ (x - a)

— ek idealized "infinitely tall, zero-width spike" of unit area jo ek genuine function nahi hai, ye point mark karta hai jahan continuous-density picture break down ho jaati hai aur variable deterministic ban jaata hai.

t = 0

par

Exp (λ)

kya karta hai?

Density apne maximum

f (0) = λ

par hoti hai aur wahan se decay karti hai; chhote waits sabse zyada probable hote hain, aur

F (0) = 0

Beta (α, β)

ka shape kaisa hoga jab dono

α, β < 1

Ek U-shape jisme mass dono endpoints

0

aur

1

par pile up hota hai aur beech mein patla ho jaata hai — central hump ka bilkul ulta (upar figure mein blue curve).

Beta (α, β)

ka shape kaisa hoga jab exactly ek of

α, β

< 1

Ek J-shape (ya reversed-J): mass us single endpoint par pile hota hai jiska exponent negative hai — near

0

agar

α < 1 < β

, near

1

agar

β < 1 < α

(upar red curve). Sirf wahi ek end blow up karta hai.

x = 0

ke paas

Gamma (α, λ)

kaisa lagta hai jab

α < 1

Density

\propto x^{α - 1}

origin par singular (unbounded) ho jaati hai,

x \to 0^{+}

par

\infty

tak shoot karti hai, phir bhi area finite rehta hai aur

1

ke barabar hota hai;

α = 1

par ye finite hoti hai (

= λ

), aur

α > 1

par

0

se shuru hokar ek hump tak rise karti hai.

x \to \pm \infty

par Normal ki tails ka limiting behaviour kya hai?

Density

\to 0

hoti hai lekin kabhi exactly zero nahi hoti; har real value ki positive density hoti hai, isliye Normal technically infinite support rakhta hai.

Jab Gamma ka shape

α \to \infty

(

λ

fixed ke saath), distribution kaisi dikhti hai?

Ye increasingly bell-shaped aur symmetric hoti jaati hai, Normal approach karti hai — ye Central Limit Theorem ka direct echo hai, kyunki ye kai exponentials ka sum hai. Dekho Central Limit Theorem.

Kisi bhi continuous distribution aur kisi bhi exact value

c

ke liye

P (X = c)

kya hoga?

Ye ==

0

== hoga, kyunki ek single point ki width zero hai aur

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

; sirf intervals positive probability carry karte hain.

Jab

α = β \to \infty

toh Beta mean ka kya hota hai?

Mean

α / (α + β) = 1/2

par fixed rehta hai, lekin variance

\to 0

, toh distribution ==tightly

0.5

ke aas paas concentrate ho jaati hai==.

Recall Har trap ki one-line summary

Density $\neq =$ probability; rate $\neq =$ mean; formula mein $σ \neq = σ^{2}$ ; tables padhne se pehle standardize karo; uniform ka variance $12$ use karta hai; single points ki probability zero hoti hai; count karo kitne parameters $1$ se below hain yeh jaanne ke liye ki kitne endpoints blow up karte hain.

Connections

Parent: 4.9.8 Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta
Poisson Process · Central Limit Theorem · Gamma Function · Beta Function · Moment Generating Functions · Conjugate Priors in Bayesian Inference · Discrete Distributions