Discrete random variables — PMF, CDF: Question bank

4.9.3 · D5 · Maths › Probability Theory & Statistics › Discrete random variables — PMF, CDF

Ek quick vocabulary refresher taaki yahan har symbol earned lage:

True or false — justify

Har

p_{X}

value 0 aur 1 ke beech (inclusive) honi chahiye.

True — har

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

ek probability hai, isliye

0 \leq p_{X} (x) \leq 1

; upper bound isliye hai kyunki ek single point ki mass kabhi bhi total mass

1

se zyada nahi ho sakti.

Agar ek function

g (x) \geq 0

satisfy karta hai aur ek countable set par

\sum_{x} g (x) = 1

hai, to woh ek valid PMF hai.

True — woh do conditions hi poora rulebook hain; ek legal PMF ke liye koi extra structure (monotonicity, smoothness) required nahi hai.

Ek PMF ek single value par

1

ke barabar ho sakta hai.

True — ek degenerate RV jo hamesha ek value

c

leta hai, uska

p_{X} (c) = 1

hoga aur baaki jagah

0

; yeh dono axioms follow karta hai aur ek bilkul legal (bhale hi boring) PMF hai.

Ek discrete RV ke liye,

P (X = x)

infinitely many values ke liye positive ho sakta hai.

True — value set countably infinite ho sakta hai (jaise Poisson

0, 1, 2, \dots

par); masses ko sirf

1

mein sum karna hota hai, jo ek infinite series kar sakti hai.

Ek CDF kisi interval par decrease kar sakta hai agar PMF mein ek "negative bump" ho.

False — PMF values kabhi negative nahi hoti, isliye cumulative sum sirf non-negative mass add karta hai; CDF har jagah non-decreasing hai, koi exception nahi.

F_{X} (x)

bahut bade

x

ke liye

1

se zyada ho sakta hai.

False — total mass exactly

1

hai, isliye jab ek baar har point collect ho jaata hai to running total

1

par saturate ho jaata hai aur wahi rehta hai;

F_{X} (+ \infty) = 1

ek ceiling hai.

Ek discrete RV ke liye,

F_{X}

continuous hota hai.

False — yeh

p_{X} (x_{i})

size ke genuine jumps wala ek step function hai; yeh sirf right-continuous hota hai, aur ek smooth curve yeh hide kar dega ki mass actually kahan baith rahi hai.

P (a < X \leq b)

aur

P (a \leq X \leq b)

hamesha same number dete hain.

False — yeh exactly

p_{X} (a)

se differ karte hain, jo left endpoint par mass hai; ek discrete RV ke liye woh mass aksar positive hoti hai, isliye bracket type matter karta hai.

Agar

F_{X} (3) = F_{X} (2.9)

hai, to

X

(2.9, 3]

mein kisi bhi value par mass nahi rakhta.

True — barabar cumulative totals ka matlab hai

2.9

aur

3

ke beech kuch naya scoop nahi hua, isliye

P (2.9 < X \leq 3) = 0

; khaaskar

p_{X} (3) = 0

CDF mein sabse bada jump sabse zyada probable value par hota hai.

True —

x_{i}

par jump height hi

p_{X} (x_{i})

hai, isliye staircase ka sabse uncha riser mode ko mark karta hai (woh value jis par sabse zyada mass hai).

Spot the error

P (X = x)

histogram bar ka area hai, isliye wider bar ka matlab zyada probability hai."

Error yeh hai ki continuous-density intuition import ki ja rahi hai; ek discrete RV ke liye probability bar ki height hai (mass khud), aur bar width ek cosmetic drawing choice hai jiska koi meaning nahi.

P (2 < X \leq 5) = F_{X} (5) - F_{X} (5) = 0

Dono terms

F_{X} (5)

nahi hone chahiye; sahi expression

F_{X} (5) - F_{X} (2)

hai, lower endpoint par cumulative total subtract karo taaki sirf

(2, 5]

mein mass bachey.

"Maine

p_{X} (4)

F_{X} (4^{-}) - F_{X} (4)

ke roop mein recover kiya."

Subtraction ulta hai; ek jump upar jaata hai, isliye

p_{X} (4) = F_{X} (4) - F_{X} (4^{-})

— value-at-the-point minus value-just-to-the-left, jo non-negative hai.

"Is function ka

\sum_{x} g (x) = 1

hai, isliye yeh valid PMF hai" — lekin ek value

g (x_{0}) = - 0.2

hai.

1

mein sum karna zaroori hai par kaafi nahi; non-negativity condition

g (x) \geq 0

violated hai, isliye negative "mass" ise turant disqualify kar deta hai.

"CDF sabse badi value par

1

tak pahunchta hai, isliye

F_{X}

ko kisi finite

x

par

1

reach karna hi hoga."

Yeh tabhi true hai jab value set finite ho; ek countably infinite RV (jaise geometric) ke liye

F_{X}

limit mein

1

approach karta hai aur literally kabhi kisi finite value par

1

nahi hota.

"Kyunki

F_{X}

right-continuous hai, isliye yeh har value par left-continuous bhi hona chahiye."

Ek jump

F_{X}

ko left se discontinuous banata hai:

F_{X} (x_{i}^{-}) < F_{X} (x_{i})

; right-continuity ka matlab hai ki closed dot (jump ki landing) right side se belong karta hai, yeh nahi ki dono sides match karti hain.

"Kisi bhi RV ke liye,

P (X < b) = P (X \leq b)

kyunki

b

ek single point hai jiska probability zero hai."

Yeh reasoning continuous RVs ke liye hold karta hai; ek discrete RV ke liye

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

positive ho sakta hai, isliye

P (X < b) = F_{X} (b) - p_{X} (b)

, jo strictly less hoga jab

b

par mass ho.

Why questions

"Add the point masses" recipe discrete RVs ke liye kyon kaam karta hai lekin continuous ones ke liye nahi?

Kyunki discrete values countable hain aur events

{X = x_{i}}

disjoint hain, countable additivity humein unki masses sum karne deta hai; ek continuous RV ki uncountably many values hoti hain jिनका har ek mass

0

hota hai, isliye summing

0

deta hai aur hume density integrate karni padti hai.

Ek valid PMF exactly

1

mein sum kyon karna chahiye, sirf kuch finite mein nahi?

Events

{X = x_{i}}

disjoint hain aur mil kar poore sample space

Ω

ko cover karte hain, isliye unki probabilities axioms se

P (Ω) = 1

mein add hoti hain — isse kam ka matlab hoga ki kuch outcomes ka koi probability nahi.

CDF ko

\leq

(yaani

P (X \leq x)

) ke saath kyun define kiya jaata hai, na ki

<

ke saath?

Convention jump ko uski value se belong karata hai, right-continuity deta hai aur clean recovery rule

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

;

<

use karne se CDF left-continuous ho jaata aur har endpoint bookkeeping shift ho jaati.

Hum poora PMF sirf CDF se kyun recover kar sakte hain?

Saari probability mass staircase ki jump heights ke roop mein dikhti hai, aur jumps ke beech CDF flat hai; har jump

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

measure karna har value ke liye

p_{X} (x_{i})

read off karta hai.

Bracket type (

<

\leq

) discrete RVs ke liye continuous se itna zyada kyun matter karta hai?

Discrete RVs ke liye ek single point positive mass

p_{X} (a)

carry kar sakta hai, isliye

a

ko include ya exclude karna answer ko us amount se change karta hai; continuous RVs ke liye har point ka mass

0

hai, jis se bracket cosmetic ban jaata hai.

CDF non-decreasing kyun hai jab ki underlying PMF upar-neeche wiggle karta hai?

CDF ek cumulative total hai; dahine move karne par sirf aur non-negative mass append ho sakta hai, kabhi remove nahi ho sakta, isliye running sum kabhi nahi gir sakta chahe individual masses upar-neeche hon.

Edge cases

x

ke liye

F_{X} (x)

kya hai jo

X

ki har possible value se chhota ho?

Yeh

0

hai — tumne abhi tak koi bhi mass-carrying value cross nahi ki, isliye running total (tumhara bucket) abhi khali hai; formally

F_{X} (- \infty) = 0

Ek degenerate RV ka CDF kaisa dikhta hai jo hamesha

c

ke barabar ho?

Ek single step:

x < c

ke liye

F_{X} (x) = 0

aur

x \geq c

ke liye

F_{X} (x) = 1

c

par height

1

ka ek jump kyunki saari mass usi ek value par baithe hai.

Agar range mein kisi value

x

ke liye

p_{X} (x) = 0

ho, to wahan CDF ka kya hota hai?

Kuch bhi visible nahi hota — staircase us

x

par bina kisi jump ke flat rehta hai, kyunki zero jump koi mass add nahi karta; woh value simply support mein nahi hai.

Kya ek discrete RV negative ya non-integer values le sakta hai, aur kya woh PMF/CDF machinery ko tod deta hai?

Haan le sakta hai (jaise values

- 2, 0.5, π

) aur kuch nahi toot-ta; machinery ko sirf value set ko countable hona chahiye, integer ya positive nahi.

Ek countably infinite RV ke liye, kya koi "last" jump hai jo CDF ko exactly

1

par land karta hai?

Nahi — jumps arbitrarily small hote jaate hain aur

F_{X} (x) \to 1

sirf limit

x \to + \infty

mein hota hai; koi final value nahi hai jahan woh

1

par snap kare, finite case ke unlike.

Ek jump value

x_{i}

par, CDF actually kaun si value leta hai — riser ka bottom ya top?

Top (right par closed dot), kyunki

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

mein

x_{i}

par mass include hai; bottom left-limit

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

hai, jise CDF pehle hi peeche chhod chuka hai.

P (X = x)

kya hai us value par jahan CDF flat hai (koi jump nahi)?

Zero — flat stretch ka matlab hai wahan koi mass add nahi hua, aur

p_{X} (x) = F_{X} (x) - F_{X} (x^{-}) = 0

; us value par koi probability nahi hai.

Connections

Probability Axioms — non-negativity aur countable additivity yahan har "true/false" verdict ko underwrite karte hain.
Continuous random variables — PDF, CDF — point-mass-vs-density contrast zyaatar bracket-type traps ka source hai.
Expectation and Variance of Discrete RVs — usi PMF par built hai jिसकी validity yeh questions probe karti hai.
Binomial Distribution, Poisson Distribution — edge cases ke peeche finite aur countably-infinite examples.
Conditional Probability — conditional PMFs upar test ki gayi har property inherit karte hain.