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Let p = 1. In this case, Sn as defined by n Sn = a + X₂ i=1 is the simple symmetric random walk. Stirling's approximation states that n! ~ n²e " √2πn (1) " -n πη Suppose a = 0. Write an exact expression for P(Sn = b), where b € {0, ±1, ±2, … }. Use Stirling's approximation to demonstrate that when n is large, P(Sn = b) as n. exp 6² 2n

Let p = 1. In this case, Sn as defined by n Sn = a + X₂ i=1 is the simple symmetric random walk. Stirling's approximation states that n! ~ n²e " √2πn (1) " -n πη Suppose a = 0. Write an exact expression for P(Sn = b), where b € {0, ±1, ±2, … }. Use Stirling's approximation to demonstrate that when n is large, P(Sn = b) as n. exp 6² 2n

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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Let \( p = \frac{1}{2} \). In this case, \( S_n \) as defined by \[ S_n = a + \sum_{i=1}^{n} X_i \] is the simple symmetric random walk. Stirling’s approximation states that \[ n! \sim n^n e^{-n} \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \quad \text{as } n \to \infty. \] Suppose \( a = 0 \). Write an exact expression for \( P(S_n = b) \), where \( b \in \{0, \pm 1, \pm 2, \ldots \} \). Use Stirling’s approximation to demonstrate that when \( n \) is large, \[ P(S_n = b) \sim \sqrt{\frac{2}{\pi n}} \exp \left( -\frac{b^2}{2n} \right). \]
Transcribed Image Text:Let \( p = \frac{1}{2} \). In this case, \( S_n \) as defined by \[ S_n = a + \sum_{i=1}^{n} X_i \] is the simple symmetric random walk. Stirling’s approximation states that \[ n! \sim n^n e^{-n} \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \quad \text{as } n \to \infty. \] Suppose \( a = 0 \). Write an exact expression for \( P(S_n = b) \), where \( b \in \{0, \pm 1, \pm 2, \ldots \} \). Use Stirling’s approximation to demonstrate that when \( n \) is large, \[ P(S_n = b) \sim \sqrt{\frac{2}{\pi n}} \exp \left( -\frac{b^2}{2n} \right). \]
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