Question

For (a) it should be s^6 instead of s^7, thank you!!

3. Find the distributions that have the generating functions (a) G(s)=(xl-s')/(1-s) (b) G(s-q+ ps.

Answer 1

(b) Bernouli distribution has generating function

Let X be a discrete random variable with Bernouli distribution with parameter p

Then, p.g.f of X is:

G(s)= q+ps

where q=1-p

Proof:

G(s)=∑ Px(x)s^x

From the definition:

Px(x) = p : x = a

1-p : x=b

so, G(s) = px(0) s^0+ px(1)s^1

=1-p+ps

=q+ps