Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's approach each part of the question in detail.
### (a) Likelihood functions \( P_{N \mid H_0}(n) \) and \( P_{N \mid H_1}(n) \)
Given that under hypothesis \( H_0 \), \( N \) follows a Poisson distribution with mean \( \lambda_0 = 4 \), and under hypothesis \( H_1 \), \( N \) follows a Poisson distribution with mean \( \lambda_1 = 6 \):
The Poisson probability mass function (PMF) is given by:
[tex]\[ P(N=n) = \frac{\lambda^n e^{-\lambda}}{n!} \][/tex]
For \( H_0 \) with \( \lambda_0 = 4 \):
[tex]\[ P_{N \mid H_0}(n) = \frac{4^n e^{-4}}{n!} \][/tex]
For \( H_1 \) with \( \lambda_1 = 6 \):
[tex]\[ P_{N \mid H_1}(n) = \frac{6^n e^{-6}}{n!} \][/tex]
Given the example value for \( N \), let \( N = 5 \):
[tex]\[ P_{N \mid H_0}(5) = \frac{4^5 e^{-4}}{5!} = 0.1563 \][/tex]
[tex]\[ P_{N \mid H_1}(5) = \frac{6^5 e^{-6}}{5!} = 0.1606 \][/tex]
### (b) Maximum a posteriori probability (MAP) hypothesis test
To design a MAP hypothesis test, we need to compare the posterior probabilities. Bayes' theorem helps us compute these probabilities.
The prior probabilities are:
[tex]\[ P(H_0) = 0.2 \][/tex]
[tex]\[ P(H_1) = 0.8 \][/tex]
The posterior probabilities are:
[tex]\[ \text{posterior}_{H_0} = P_{N \mid H_0}(5) \times P(H_0) \][/tex]
[tex]\[ \text{posterior}_{H_1} = P_{N \mid H_1}(5) \times P(H_1) \][/tex]
Calculating the posterior probabilities:
[tex]\[ \text{posterior}_{H_0} = 0.1563 \times 0.2 = 0.0313 \][/tex]
[tex]\[ \text{posterior}_{H_1} = 0.1606 \times 0.8 = 0.1285 \][/tex]
Comparing the posterior probabilities:
[tex]\[ \text{posterior}_{H_0} = 0.0313 \][/tex]
[tex]\[ \text{posterior}_{H_1} = 0.1285 \][/tex]
Since \( \text{posterior}_{H_1} \) is greater than \( \text{posterior}_{H_0} \), the MAP decision is to choose \( H_1 \).
### (c) Total error probability \( P_{\text{ERR}} \) of the hypothesis test
The total error probability \( P_{\text{ERR}} \) includes both Type I and Type II errors.
- Type I error is the probability of deciding \( H_1 \) when \( H_0 \) is true.
- Type II error is the probability of deciding \( H_0 \) when \( H_1 \) is true.
Given that:
[tex]\[ P_{\text{Type I error}} = P(H_0) \times \left(1 - P_{N \mid H_0}(5)\right) \][/tex]
[tex]\[ P_{\text{Type II error}} = P(H_1) \times \left(1 - P_{N \mid H_1}(5)\right) \][/tex]
Calculating these probabilities:
[tex]\[ P_{\text{Type I error}} = 0.2 \times (1 - 0.1563) = 0.1687 \][/tex]
[tex]\[ P_{\text{Type II error}} = 0.8 \times (1 - 0.1606) = 0.6715 \][/tex]
So, the total error probability:
[tex]\[ P_{\text{ERR}} = P_{\text{Type I error}} + P_{\text{Type II error}} \][/tex]
[tex]\[ P_{\text{ERR}} = 0.1687 + 0.6715 = 0.8402 \][/tex]
### Summary:
(a) \( P_{N \mid H_0}(5) = 0.1563 \) \\
\( P_{N \mid H_1}(5) = 0.1606 \)
(b) MAP decision is to choose \( H_1 \).
(c) The total error probability [tex]\( P_{\text{ERR}} \)[/tex] is 0.8402.
### (a) Likelihood functions \( P_{N \mid H_0}(n) \) and \( P_{N \mid H_1}(n) \)
Given that under hypothesis \( H_0 \), \( N \) follows a Poisson distribution with mean \( \lambda_0 = 4 \), and under hypothesis \( H_1 \), \( N \) follows a Poisson distribution with mean \( \lambda_1 = 6 \):
The Poisson probability mass function (PMF) is given by:
[tex]\[ P(N=n) = \frac{\lambda^n e^{-\lambda}}{n!} \][/tex]
For \( H_0 \) with \( \lambda_0 = 4 \):
[tex]\[ P_{N \mid H_0}(n) = \frac{4^n e^{-4}}{n!} \][/tex]
For \( H_1 \) with \( \lambda_1 = 6 \):
[tex]\[ P_{N \mid H_1}(n) = \frac{6^n e^{-6}}{n!} \][/tex]
Given the example value for \( N \), let \( N = 5 \):
[tex]\[ P_{N \mid H_0}(5) = \frac{4^5 e^{-4}}{5!} = 0.1563 \][/tex]
[tex]\[ P_{N \mid H_1}(5) = \frac{6^5 e^{-6}}{5!} = 0.1606 \][/tex]
### (b) Maximum a posteriori probability (MAP) hypothesis test
To design a MAP hypothesis test, we need to compare the posterior probabilities. Bayes' theorem helps us compute these probabilities.
The prior probabilities are:
[tex]\[ P(H_0) = 0.2 \][/tex]
[tex]\[ P(H_1) = 0.8 \][/tex]
The posterior probabilities are:
[tex]\[ \text{posterior}_{H_0} = P_{N \mid H_0}(5) \times P(H_0) \][/tex]
[tex]\[ \text{posterior}_{H_1} = P_{N \mid H_1}(5) \times P(H_1) \][/tex]
Calculating the posterior probabilities:
[tex]\[ \text{posterior}_{H_0} = 0.1563 \times 0.2 = 0.0313 \][/tex]
[tex]\[ \text{posterior}_{H_1} = 0.1606 \times 0.8 = 0.1285 \][/tex]
Comparing the posterior probabilities:
[tex]\[ \text{posterior}_{H_0} = 0.0313 \][/tex]
[tex]\[ \text{posterior}_{H_1} = 0.1285 \][/tex]
Since \( \text{posterior}_{H_1} \) is greater than \( \text{posterior}_{H_0} \), the MAP decision is to choose \( H_1 \).
### (c) Total error probability \( P_{\text{ERR}} \) of the hypothesis test
The total error probability \( P_{\text{ERR}} \) includes both Type I and Type II errors.
- Type I error is the probability of deciding \( H_1 \) when \( H_0 \) is true.
- Type II error is the probability of deciding \( H_0 \) when \( H_1 \) is true.
Given that:
[tex]\[ P_{\text{Type I error}} = P(H_0) \times \left(1 - P_{N \mid H_0}(5)\right) \][/tex]
[tex]\[ P_{\text{Type II error}} = P(H_1) \times \left(1 - P_{N \mid H_1}(5)\right) \][/tex]
Calculating these probabilities:
[tex]\[ P_{\text{Type I error}} = 0.2 \times (1 - 0.1563) = 0.1687 \][/tex]
[tex]\[ P_{\text{Type II error}} = 0.8 \times (1 - 0.1606) = 0.6715 \][/tex]
So, the total error probability:
[tex]\[ P_{\text{ERR}} = P_{\text{Type I error}} + P_{\text{Type II error}} \][/tex]
[tex]\[ P_{\text{ERR}} = 0.1687 + 0.6715 = 0.8402 \][/tex]
### Summary:
(a) \( P_{N \mid H_0}(5) = 0.1563 \) \\
\( P_{N \mid H_1}(5) = 0.1606 \)
(b) MAP decision is to choose \( H_1 \).
(c) The total error probability [tex]\( P_{\text{ERR}} \)[/tex] is 0.8402.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
