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Discrete Random Variable
LetY be a random variable

Ry ≡ Range of Y
≡ set of all possible values of Y
≡ finite or countable(countably infinite)
The p.m.f(probability mass function) of Y
if Ry = y1 , y2 , . . . , yk i.e Ry is finite.

Y

y1

y2

y3

PY (y)
,
,
PY (y3 )
PY (y) = P (Y = y)

...

yk

...

PY (yk )

they must have

(a)PY (y) ≥ 0
(b)

k
X

PY (yk ) = 1

j=1

Example 1
Discrete uniform on counting number
Y ∼ uniform(1,. . .,n)
Y follows a discrete uniform distribution on 1, 2, . . . , n
step 1→ RY = [1, 2, . . . , n]
step 2 → pmf of Y is given by;
PY (y) =

1
n

∀y ∈ 1, . . . , n

Y

1

2

...

n

PY (y)

1
n

1
n

...

1
n

Example 2
RY = [−2, −1, 0, 3, 7]
PY (y) = (y 2 + 1)c
Find c such that PY (y) is a valid p.m.f.

1

solution
PY (y)=c[5+2+1+10+48]
c[33]=1
1
33

c=
Pk

j=1

PY (yk )=1

=PY (−2) + PY (−1) + PY (0) + PY (3) + PY (7)=1
c(68)=1
1
c= 68
RY = [−2, −1, 0, 3, 7]

PY (y) = P (Y = y)
1
= 68
(y 2 + 1)

PY (y) = P (Y = y)=



 1 (y 2 + 1), if y ∈ Ry
68


0,

1

otherwise.

Example 3

let y=




1

if outcomeis”success”



o

if outcomeis”f ailure”

=1 (w is ”success”)
RY = [0, 1]
PY (y) = P (Y = y) = P y (1 − P )1−y
where P=P(w is ”success”)
Y

0

1

PY (y)

1-P

P

Example 4
Binomial Distribution
Y ≡ Number of ”success” out of n independent Bernoulli trials.
RY = [0, 1, 2, 3, . . . , n]
Pn
Y = i=1 λ1(wi is ”success”)
pmf of Y, PY (y) = P(Y = y) =

n!
Py (1 − P)n−y
y!(n − y)!

2

Cumulative Distribution(C.D.F)
Let Y be a discrete random variable on RY with pmf

Y

(y)

then the cdf of Y is given by;

FY (y) = P(Y ≤ y) = PY (y1 ) + PY (y2 ) + . . . + PY (yn ) =

a≤y
X

PY (a)

Example 5
Y ∼ Bin(3, 21 )
write down;

(a) RY = [0, 1, 2, 3]
(b) PY (y) =

3!
3!
( 1 )y ( 1 )3−y =
( 1 )3
y!(3 − y)! 2 2
y!(3 − y)! 2

(c) FY (y) is table format
(d) FY (y) in compact algebraic form

Solution
(a)

(c)

Y

0

1

2

3

PY (y)

1
8

3
8

3
8

1
8

Y

0

1

2

3

FY (y)

1
8

1
2

7
8

1

(d) FY (y) =

P

a≤y

PY (a)

* Given FY (y), it is often very desired to obtain quantiles of inverse cdf numbers
*Definition
Given α ∈ (0, 1]
FY−1 (α) = inf[y ∈ RY : FY (y) ≡ P(Y ≤ y) = α]
If Y ∼ Bin(3, 21 )
then FY−1 ( 87 ) = 2

3

Question
Fi...