Can anyone help with part c and d Let the time until a claim

Let the time until a claim N is paid have the geometric distribution P(N = n) = p(l - p)^n, n = 0,1,2,.... What is P(N n), n = 0,1,2,...? What is the (discrete) hazard P(N = n|N n), n = 0,1,2,...? What is E[N]? Assume the payout is a known, not random, amount 1, but that future payouts are discounted with a factor v

Solution

part c)

E[v^N]=v⁰p(1-p)⁰+v¹p(1-p)¹+v²p(1-p)²+v³p(1-p)³+...............

        =p+vp(1-p)+v²p(1-p)²+v³p(1-p)³+...............

        =p[1+v(1-p)+v²(1-p)²+v³(1-p)³+.....................]

        =p(1/{1-v(1-p)})    [answer]

d) though the language of the problem is not cleared but it may be that we need to find E[v^N+1] as payout is starting from next year.

so E[v^N+1]=v¹p(1-p)⁰+v²p(1-p)¹+v³p(1-p)²+v⁴p(1-p)³+...............

               =vp[1+v(1-p)+v²(1-p)²+v³(1-p)³+...............]

               =vp(1/{1-v(1-p)})   [answer]