Question

QUESTION 2 A poisonous gas sensor measures a continuous random variable X. In the presence of toxic gases, hypothesis H1, X is distributed as e 16 x > 0 fx|h1(2|H1) x < 0 When there are no toxic gases present (Hypothesis HO), X is distributed as [x170(0|40) = {* * *23 x 20 x < 0 Find the Maximum Likelihood detector for this problem, and express it as a comparison of the measurement X to a threshold (remember that X can only take nonnegative values). Enter the value of the threshold up to two decimal places:

Answer 1

fxh.) e The likelihood ratio = A(z) = 16 *) fx/H.(z) - Zen We reject Ho at level a = 0.05 if A(z) <c or, z? <c or < <k Critical region for likelihood ratio test at level 0.05 is{X:X<k} where k is such that Pho(X < k) = 0,05 or, 1 e 4dr = 0.05 or, [-2-3)* = 0.05 or, 1-4-4 = 0.05 or, e-4 = 0.95 or, t = 0.0513 or, k = V2*0.0513 = 0.32 Answer: 0.32.