pgf-hw - X n =3 †z 2  n-2 = z 2 ∞ X n =1 †z 2  n =...

A few basic questions on probability generating functions
1 Let X be a random variable with the following mass function:
P [ X = - 2] =
1
3
P [ X = 3] =
1
6
P [ X = π ] =
1
8
P
•
X =
7
2
‚
=
3
8
.
Find the probability generating function G
X
( z ).
2 Suppose P [ X = n ] =
1
2 n - 2
for n = 3 , 4 , 5 , . . . . Find a compact formula for the probability
generating function G
X
( z ).
3 Suppose G
X
( z ) =
(
1
3
z +
2
3
)
4
z
.
What is the range of X ? What is the mass function?

A few basic questions on probability generating functions Page 2 of 2
Solutions
(1) Remember that your output values become powers on z , and your probabilities become
coefficients of those powers of z .
G
X
( z ) =
1
3
z
- 1
+
1
6
z
3
+
1
8
z
π
+
3
8
z
7 / 2
.
(2) By definition,
G
X
( z ) =
X
n ∈ Range ( X )
P [ X = n ] z
n
.
Here, we have the range and also the probabilities, so
G
X
( z ) =
∞ X
n =3
1
2 n - 2
z
n
.
Factoring out a z
2
to even up the powers inside, we get