1. The random variable z describes the observation of a physical phenomenon whose statistical description depends of one of two hypotheses. Conditioned on hypothesis H₁, the random variable z is Erlang (3,0.1), while conditioned on hypothesis Ho, the random variable z is Erlang (1,0.1). If Pr{H} = 0.65, and Pr{H} = 0.35, a) develop the likelihood ratio to implement a MAP decision test in order to decide between the two hypotheses. b) Obtain the probability of error of the test. c) If z=15.5, what decision results from the test?
1. The random variable z describes the observation of a physical phenomenon whose statistical description depends of one of two hypotheses. Conditioned on hypothesis H₁, the random variable z is Erlang (3,0.1), while conditioned on hypothesis Ho, the random variable z is Erlang (1,0.1). If Pr{H} = 0.65, and Pr{H} = 0.35, a) develop the likelihood ratio to implement a MAP decision test in order to decide between the two hypotheses. b) Obtain the probability of error of the test. c) If z=15.5, what decision results from the test?
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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![**Problem Description:**
1. The random variable \( z \) describes the observation of a physical phenomenon whose statistical description depends on one of two hypotheses. Conditioned on hypothesis \( H_1 \), the random variable \( z \) follows an Erlang distribution with parameters \( (3, 0.1) \). Conditioned on hypothesis \( H_0 \), the random variable \( z \) follows an Erlang distribution with parameters \( (1, 0.1) \).
The prior probabilities are given as:
\[
\Pr\{H_1\} = 0.65, \quad \Pr\{H_0\} = 0.35
\]
Tasks:
a) Develop the likelihood ratio to implement a MAP (Maximum A Posteriori) decision test in order to decide between the two hypotheses.
b) Obtain the probability of error of the test.
c) If \( z = 15.5 \), determine the decision result from the test.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9328926a-b343-4d47-8a00-cbd9e6e36ac2%2F099a2538-8a50-4865-9b93-4689e59b1649%2Fq7e0jbt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
1. The random variable \( z \) describes the observation of a physical phenomenon whose statistical description depends on one of two hypotheses. Conditioned on hypothesis \( H_1 \), the random variable \( z \) follows an Erlang distribution with parameters \( (3, 0.1) \). Conditioned on hypothesis \( H_0 \), the random variable \( z \) follows an Erlang distribution with parameters \( (1, 0.1) \).
The prior probabilities are given as:
\[
\Pr\{H_1\} = 0.65, \quad \Pr\{H_0\} = 0.35
\]
Tasks:
a) Develop the likelihood ratio to implement a MAP (Maximum A Posteriori) decision test in order to decide between the two hypotheses.
b) Obtain the probability of error of the test.
c) If \( z = 15.5 \), determine the decision result from the test.
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