Solution:
P(W=w)=h(w)=nCwpw(1−p)n−w
(a) h(w−1)=nCw−1pw−1(1−p)n−w+1
Now, h(w−1)h(w)=nCw−1pw−1(1−p)n−w+1nCwpw(1−p)n−w
=(w−1)!(n−w+1)n!(1−p)w!(n−w)!n!p
=(w!)(n−w)!(1−p)(w−1)!(n−w+1)!p=w(w−1)!(n−w)!(1−p)(w−1)!(n−w+1)(n−w)!p
=w(1−p)(n−w+1)p
(b) E[W(W−1)]=E[W2−W]=E(W2)−E(W) ...(i)
Now, E(W)=np,Var(W)=E(W2)−[E(W)]2=np(1−p)
So, E(W2)=[E(W)]2+np(1−p)=n2p2+np(1−p)
Putting these values in (i)
E[W(W−1)]=E(W2)−E(W)=n2p2+np(1−p)−np
=n2p2+np−np2−np=n2p2−np2=np2(n−1)=n(n−1)p2
(c) E[W+11]=Σ(w+11)P(W=w)
=Σw=0n(w+11)×nCwpw(1−p)n−w
=p(n+1)1Σw=0n.n+1Cw+1.pw−1(1−p)n−w
=p(n+1)1.(1−(1−p)n+1)
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