Simple Stochastic Optimization Problem

I'm completely new to the stochastic optimization and struggling with the following problem: Suppose that Θ contains only two options, Θ = {θ1, θ2}, and that L(θ1) = 0 and L(θ2) = 1 (so θ1 is the optimal θ since it has the lowest value of L). Suppose that you do not know the values of L and can only collect one noisy measurement y(θi ) at each of the two θi to try to determine which θi is “better” (lower) relative to L (i.e., you only observe the noisy measurements y(θi ) = L(θi ) + εi for i = 1 and 2). Further, suppose that the noise terms ε1 and ε2 for the two measurements are independent of each other and take on values {−2, 0, 2} with probabilities 2/5, 1/5, and 2/5, respectively. What is the probability of incorrectly selecting θ2 as the optimal θ when the choice is based on which θi has the lowest y(θi )? I started with calculating y(theta) for each value of noise, but don't know how to proceed next.

Thank you for any contribution!

• Welcome to mathematics SE. What have you tried? People here like to see the effort made. Nov 12, 2019 at 15:59
• Thank you! I am thinking about comparing probabilities of y theta as fractions.. other then that, I do not know how to proceed. That's what I have done: y(theta1) with e1= -2; with e2= 0 eith e3= 2. y(theta2) with e1= -2, with e2=0, with e3=3. lowest y(theta)= -2 Nov 12, 2019 at 16:12
• there is no optimization involved here, it is simply about enumerating all possibilities and their probabilities Nov 12, 2019 at 18:09