Problem Statement:
For this problem the observation Y = (Y1,Y2) is a vector of two random variables Y1,Y2. The hypotheses are as follows
H0 : Y1 ∼ N(3,1),Y2 ∼ N(−1,1) H1 : Y1,Y2 ∼ Px,
where N(µ,σ2) is the Gaussian distribution with mean µ and variance σ2, and Px is the distribution with the following probability density function
, for all x ∈ R.
In both the hypotheses, the random variables Y1 and Y2 are independent of each other. Use uniform cost, i.e., C00 = C11 = 0, C10 = C01 = 1.
You must use Monte Carlo simulation to find and plot V (π0) versus π0, for π0 = 0.1,0.2,...,0.9.
Simulation Details:
• For each value π0, you must generate 106 instances of Y and apply the Bayes decision rule to perform detection. Use the average cost of these 106 instances as your estimate for V (π0).
• In each instance, you must randomly generate Y according to either H0 (with probability π0) or H1 (with probability 1 − π0).
• Ensure that the observation vectors generated for H0 and H1 indeed satisfy the probability distributions described above.
• Reference value: at π0 = 0.25, V ≈ 0.0755.
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