Answer :
Sure, let's solve each part of this problem step-by-step:
### (a) Likelihood Functions
We are given that the number of call attempts [tex]\( N \)[/tex] follows a Poisson distribution under two hypotheses [tex]\( H_0 \)[/tex] and [tex]\( H_1 \)[/tex]:
- Under [tex]\( H_0 \)[/tex]: The mean number of call attempts [tex]\( \lambda_0 \)[/tex] is 6.
- Under [tex]\( H_1 \)[/tex]: The mean number of call attempts [tex]\( \lambda_1 \)[/tex] is 8.
The Poisson probability mass function (PMF) is given by:
[tex]\[ P(N = n \mid \lambda) = \frac{e^{-\lambda} \lambda^n}{n!} \][/tex]
Thus, for each hypothesis:
[tex]\[ P(N = n \mid H_0) = \frac{e^{-6} \cdot 6^n}{n!} \][/tex]
[tex]\[ P(N = n \mid H_1) = \frac{e^{-8} \cdot 8^n}{n!} \][/tex]
### (b) MAP Hypothesis Test
The Maximum a Posteriori (MAP) hypothesis test aims to choose the hypothesis that maximizes the posterior probability. This is given by comparing the likelihoods weighted by their prior probabilities.
[tex]\[ \text{Select } H_0 \text{ if } P(N = n \mid H_0) \cdot P(H_0) > P(N = n \mid H_1) \cdot P(H_1) \][/tex]
[tex]\[ \text{Select } H_1 \text{ otherwise } \][/tex]
Given:
[tex]\[ P(H_0) = 0.3 \][/tex]
[tex]\[ P(H_1) = 0.7 \][/tex]
Thus, the decision rule is:
[tex]\[ \frac{e^{-6} \cdot 6^n}{n!} \cdot 0.3 \quad \text{vs} \quad \frac{e^{-8} \cdot 8^n}{n!} \cdot 0.7 \][/tex]
[tex]\[ \text{Select } H_0 \text{ if } 0.3 \cdot 6^n \cdot e^{-6} > 0.7 \cdot 8^n \cdot e^{-8} \][/tex]
[tex]\[ \text{Select } H_1 \text{ otherwise } \][/tex]
### (c) Total Error Probability [tex]\(P_{\text{ERR}}\)[/tex]
To compute the total error probability [tex]\(P_{\text{ERR}}\)[/tex], we need to account for the probabilities of false alarms and misses. Here, the total error probability is the sum of these probabilities weighted by their prior:
- A false alarm occurs when [tex]\( H_0 \)[/tex] is chosen but [tex]\( N \)[/tex] was actually generated by [tex]\( H_1 \)[/tex].
- A miss occurs when [tex]\( H_1 \)[/tex] is chosen but [tex]\( N \)[/tex] was actually generated by [tex]\( H_0 \)[/tex].
The total error probability [tex]\(P_{\text{ERR}}\)[/tex] is calculated as follows:
[tex]\[ P_{\text{ERR}} = \sum_{n=0}^{\infty} \left[ P(N = n \mid H_1) \cdot P(H_1) \cdot I(\text{Decision} = H_0) + P(N = n \mid H_0) \cdot P(H_0) \cdot I(\text{Decision} = H_1) \right] \][/tex]
Where [tex]\( I(\cdot) \)[/tex] is an indicator function that is 1 if the condition is true, otherwise 0. Based on the calculated result:
[tex]\[ P_{\text{ERR}} \approx 0.2842257302459429 \][/tex]
### (d) Average Cost of MAP Policy and Minimum Cost Policy
The average cost of the MAP policy is given by:
[tex]\[ \text{Average Cost}_{\text{MAP}} = P_{\text{ERR}} \cdot C_{01} + (1 - P_{\text{ERR}}) \cdot C_{10} \][/tex]
Given the costs:
[tex]\[ C_{10} = 10 \][/tex]
[tex]\[ C_{01} = 10^4 \][/tex]
Using the total error probability [tex]\( P_{\text{ERR}} = 0.2842257302459429 \)[/tex]:
[tex]\[ \text{Average Cost}_{\text{MAP}} = 0.2842257302459429 \cdot 10000 + (1 - 0.2842257302459429) \cdot 10 \][/tex]
[tex]\[ \approx 2849.4150451569694 \][/tex]
The minimum cost policy considers choosing the action with the minimum expected cost. In this case, the minimum possible cost comes just from making a decision that avoids the higher cost miss penalty and defaults to the false alarm cost:
[tex]\[ \text{Average Cost}_{\text{Minimum}} = \min(\text{Average Cost}_{\text{MAP}}, C_{01}, C_{10}) \][/tex]
[tex]\[ \text{Average Cost}_{\text{Minimum}} = 10 \][/tex]
Therefore, the total results are:
[tex]\[ P_{\text{ERR}} \approx 0.2842257302459429 \][/tex]
[tex]\[ \text{Average Cost}_{\text{MAP}} \approx 2849.4150451569694 \][/tex]
[tex]\[ \text{Average Cost}_{\text{Minimum}} = 10 \][/tex]
### (a) Likelihood Functions
We are given that the number of call attempts [tex]\( N \)[/tex] follows a Poisson distribution under two hypotheses [tex]\( H_0 \)[/tex] and [tex]\( H_1 \)[/tex]:
- Under [tex]\( H_0 \)[/tex]: The mean number of call attempts [tex]\( \lambda_0 \)[/tex] is 6.
- Under [tex]\( H_1 \)[/tex]: The mean number of call attempts [tex]\( \lambda_1 \)[/tex] is 8.
The Poisson probability mass function (PMF) is given by:
[tex]\[ P(N = n \mid \lambda) = \frac{e^{-\lambda} \lambda^n}{n!} \][/tex]
Thus, for each hypothesis:
[tex]\[ P(N = n \mid H_0) = \frac{e^{-6} \cdot 6^n}{n!} \][/tex]
[tex]\[ P(N = n \mid H_1) = \frac{e^{-8} \cdot 8^n}{n!} \][/tex]
### (b) MAP Hypothesis Test
The Maximum a Posteriori (MAP) hypothesis test aims to choose the hypothesis that maximizes the posterior probability. This is given by comparing the likelihoods weighted by their prior probabilities.
[tex]\[ \text{Select } H_0 \text{ if } P(N = n \mid H_0) \cdot P(H_0) > P(N = n \mid H_1) \cdot P(H_1) \][/tex]
[tex]\[ \text{Select } H_1 \text{ otherwise } \][/tex]
Given:
[tex]\[ P(H_0) = 0.3 \][/tex]
[tex]\[ P(H_1) = 0.7 \][/tex]
Thus, the decision rule is:
[tex]\[ \frac{e^{-6} \cdot 6^n}{n!} \cdot 0.3 \quad \text{vs} \quad \frac{e^{-8} \cdot 8^n}{n!} \cdot 0.7 \][/tex]
[tex]\[ \text{Select } H_0 \text{ if } 0.3 \cdot 6^n \cdot e^{-6} > 0.7 \cdot 8^n \cdot e^{-8} \][/tex]
[tex]\[ \text{Select } H_1 \text{ otherwise } \][/tex]
### (c) Total Error Probability [tex]\(P_{\text{ERR}}\)[/tex]
To compute the total error probability [tex]\(P_{\text{ERR}}\)[/tex], we need to account for the probabilities of false alarms and misses. Here, the total error probability is the sum of these probabilities weighted by their prior:
- A false alarm occurs when [tex]\( H_0 \)[/tex] is chosen but [tex]\( N \)[/tex] was actually generated by [tex]\( H_1 \)[/tex].
- A miss occurs when [tex]\( H_1 \)[/tex] is chosen but [tex]\( N \)[/tex] was actually generated by [tex]\( H_0 \)[/tex].
The total error probability [tex]\(P_{\text{ERR}}\)[/tex] is calculated as follows:
[tex]\[ P_{\text{ERR}} = \sum_{n=0}^{\infty} \left[ P(N = n \mid H_1) \cdot P(H_1) \cdot I(\text{Decision} = H_0) + P(N = n \mid H_0) \cdot P(H_0) \cdot I(\text{Decision} = H_1) \right] \][/tex]
Where [tex]\( I(\cdot) \)[/tex] is an indicator function that is 1 if the condition is true, otherwise 0. Based on the calculated result:
[tex]\[ P_{\text{ERR}} \approx 0.2842257302459429 \][/tex]
### (d) Average Cost of MAP Policy and Minimum Cost Policy
The average cost of the MAP policy is given by:
[tex]\[ \text{Average Cost}_{\text{MAP}} = P_{\text{ERR}} \cdot C_{01} + (1 - P_{\text{ERR}}) \cdot C_{10} \][/tex]
Given the costs:
[tex]\[ C_{10} = 10 \][/tex]
[tex]\[ C_{01} = 10^4 \][/tex]
Using the total error probability [tex]\( P_{\text{ERR}} = 0.2842257302459429 \)[/tex]:
[tex]\[ \text{Average Cost}_{\text{MAP}} = 0.2842257302459429 \cdot 10000 + (1 - 0.2842257302459429) \cdot 10 \][/tex]
[tex]\[ \approx 2849.4150451569694 \][/tex]
The minimum cost policy considers choosing the action with the minimum expected cost. In this case, the minimum possible cost comes just from making a decision that avoids the higher cost miss penalty and defaults to the false alarm cost:
[tex]\[ \text{Average Cost}_{\text{Minimum}} = \min(\text{Average Cost}_{\text{MAP}}, C_{01}, C_{10}) \][/tex]
[tex]\[ \text{Average Cost}_{\text{Minimum}} = 10 \][/tex]
Therefore, the total results are:
[tex]\[ P_{\text{ERR}} \approx 0.2842257302459429 \][/tex]
[tex]\[ \text{Average Cost}_{\text{MAP}} \approx 2849.4150451569694 \][/tex]
[tex]\[ \text{Average Cost}_{\text{Minimum}} = 10 \][/tex]
