Let θ be a bernoulli random variable that indicates which one of two hypotheses is true, and let θ=1)= . under the hypothesis θ=0 , the random variable has a normal distribution with mean 0 , and variance 1 . under the alternative hypothesis θ=1 , has a normal distribution with mean 2 and variance 1 . consider the map rule for deciding between the two hypotheses, given that = . suppose for this part of the problem that =2/3 . the map rule can choose in favor of the hypothesis θ=1 if and only if ≥ . find the value of .

Arrange the steps to solve this system of linear equations in the correct sequence. x + y = -2 2x – 3y = -9 tiles subtract 3x + 3y = -6 (obtained in step 1) from 2x – 3y = -9 (given) to solve for x. substitute the value of x in the first equation (x + y = -2) to get y = 1. the solution for the system of equations is (-3, 1). x = -15 the solution for the system of equations is (-15, 13). add 3x + 3y = -6 (obtained in step 1) to 2x – 3y = -9 (given), and solve for x. x = -3 substitute the value of x in the first equation (x + y = -2) to get y = 13. multiply the first equation by 3: 3(x + y) = 3(-2) 3x + 3y = -6.