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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)
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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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.
Transcribed Image Text:**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.
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