4.5. How can I prove these three equalities? **Mathematical Expectation of a Nonnegative Integer-Valued Random Variable****Problem 4.5:**Let \( N \) be a nonnegative integer-valued random variable. For nonnegative values \( a_j, j \geq 1 \), show that:\[\sum_{j=1}^{\infty} (a_1 + \ldots + a_j) P(N = j) = \sum_{i=1}^{\infty} a_i P(N \geq i)\]**Next, show that:**The expected value of \( N \), denoted \( E[N] \), is given by:\[E[N] = \sum_{i=1}^{\infty} P(N \geq i)\]**And, demonstrate that:**The expectation of the product \( N(N + 1) \) is given by:\[E[N(N + 1)] = 2 \sum_{i=1}^{\infty} i P(N \geq i)\]---**Explanation:**The problem explores the relationships between probability mass function, distribution function, and expectation for a nonnegative integer-valued random variable. It involves proving identities that connect summations of probabilities with expectations, which are fundamental concepts in probability and statistics.

Answered: Image /qna-images/answer/3eeba5e7-30b6-4bf2-ae0d-0d78ee49b391.jpg

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

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

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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