Let Θ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P(Θ=1)=p. Under the hypothesis Θ=0, the random variable X is uniformly distributed over the interval [0,1]. Under the alternative hypothesis Θ=1, the PDF of X is given by

Question

fX∣Θ(x∣1)={2x,0, if 0≤x≤1, otherwise.
Consider the MAP rule for deciding between the two hypotheses, given that X=x.

Suppose for this part of the problem that p=3/5. The MAP rule can choose in favor of the hypothesis Θ=1 if and only if x≥c1. Find the value of c1.

c1= - unanswered
Assume now that p is general such that 0≤p≤1. It turns out that there exists a constant c such that the MAP rule always decides in favor of the hypothesis Θ=0 if and only if p<c. Find c.

c= - unanswered
For this part of the problem, assume again that p=3/5. Find the conditional probability of error for the MAP decision rule given that the hypothesis Θ=0 is true.

P(error∣Θ=0)= - unanswered
Find the probability of error associated with the MAP rule as a function of p. Express your answer in terms of p using standard notation.

When p≤1/3, P(error)= p - unanswered
p

When p≥1/3, P(error)=

Answer 1

Could anyone please provide answer for last part?

Answer 2

Could you possibly give the answers for the first parts as well?

Answer 3

1. 1/3
2. 1/3
3. 2/3
4. p≤1/3: p

Answer 4

Any clue for the case p>= 1/3?

Answer 5

4 - b. ???

Answer 6

Any idea what the answer to last question is?

Answer 7

4b. ((1-p)*(5*p-1))/(4*p)

Answer 8

1. 0.653
2. 0.257
3. 0.159