Let X; e {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2,..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X, (p | Xn).
Let X; e {1,2, 3, ...} be the number of days until relapse for patient i who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf iid X1, X2,..., Xn fxp(x | p) = (1 – p)*-'p for 0 < p < 1 defined on x E {1,2, 3, ...} and 0 elsewhere. Here, p is the risk of relapse on each day. 1. Using a p~ Beta(a, B) prior, derive the posterior density of p, fp|X, (p | Xn).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Let \( X_i \in \{1, 2, 3, \ldots\} \) be the number of days until relapse for patient \( i \) who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf
\[
X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} f_{X \mid p}(x \mid p) = (1-p)^{x-1}p
\]
for \( 0 < p < 1 \) defined on \( x \in \{1, 2, 3, \ldots\} \) and 0 elsewhere. Here, \( p \) is the risk of relapse on each day.
1. Using a \( p \sim \text{Beta}(\alpha, \beta) \) prior, derive the posterior density of \( p \), \( f_{p \mid X_n}(p \mid X_n) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F361779f5-7517-4850-b433-f5d42195c84d%2Febd12f27-f530-40d4-9c1f-d25756984b63%2Fwblfhb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( X_i \in \{1, 2, 3, \ldots\} \) be the number of days until relapse for patient \( i \) who is diagnosed with multiple sclerosis and currently in remission. We model this data using a geometric distribution with pmf
\[
X_1, X_2, \ldots, X_n \overset{\text{iid}}{\sim} f_{X \mid p}(x \mid p) = (1-p)^{x-1}p
\]
for \( 0 < p < 1 \) defined on \( x \in \{1, 2, 3, \ldots\} \) and 0 elsewhere. Here, \( p \) is the risk of relapse on each day.
1. Using a \( p \sim \text{Beta}(\alpha, \beta) \) prior, derive the posterior density of \( p \), \( f_{p \mid X_n}(p \mid X_n) \).
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