4.5. Let N be a nonnegative integer-valued random variable. For nonnegative values aj, j ≥ 1, show that Then show that and ∞ j = 1 (a₁ + ... + a₁)P(N = j} = a;P[N ≥ i} i = 1 E[N] = Σ i=1 E[N(N+1)] = 2 P{N ≥ i} 2 i 1 iP {N ≥ i} 4
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4.5. How can I prove these three equalities?
![4.5. Let N be a nonnegative integer-valued random variable. For nonnegative
values aj, j ≥ 1, show that
Then show that
and
Σ (a₁ +
Nil
j = 1
(a₁ + ... + a;)P{N = j} =
E[N] Σ P[N = i}
E[N(N+1)] = 2
Σ a₁P{N > i}
i = 1
i = 1
iP (N≥ i)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0618968-6b5a-406b-a92d-661b061825b7%2F3eeba5e7-30b6-4bf2-ae0d-0d78ee49b391%2Fjr6g4oyj_processed.jpeg&w=3840&q=75)
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