Identically distributed gaussian random variables with zero

Question

Question: 1. Derive a sequential ratio test when the observations are

H₁: Z_k = Y_k¹        K = 1, 2,. . .

H₀: Z_k = Y_k⁰        K = 1, 2,. . .

where and are independent, identically distributed Gaussian random variables with zero mean and variances and , respectively. Assume that σ1 > σ0 and that PF = 0.2 and PM = 0.1. Find the average number of samples for the test to terminate if σ0 = 1,σ1 = 2, and P(H0) = P(H1) = ½.

2. Consider the binary hypothesis-testing problem with

H₀: Z_k = 1 + V_k     k = 1, 2,. . .

H₁: Z_k = -1 + V_k     k = 1, 2,. . .

Obtain a SPRT for this case if Vk is zero mean, white, and Gaussian with variance σ2.