If Ho is the true hypothesis, the random variable X takes on values 0. 1, 2, and
ID: 2978843 • Letter: I
Question
If Ho is the true hypothesis, the random variable X takes on values 0. 1, 2, and 3 with probabilities 0.1. 0.2, 0.3, and 0.4 respectively. If H1 is the true hypothesis, the random variable X takes on values 0, 1, 2, and 3 with probabilities 0.4, 0.3, 0.2, and 0.1 respectively. Find the likelihood matrix L and indicate the maximum-likelihood decision rule by shading the appropriate entries in L. What is the false-alarm probability PFA and what is the missed-detection probability PMD for the maximum-likelihood decision rule? Now suppose that the hypotheses have a priori probabilities pi o = 0.7 and pi 1 = 0.3. Use the law of total probability to find the average error probability of the maximum-likelihood decision rule that you found in part (a). Use the a priori probabilities given in part (b) to find the joint probability matrix J and indicate on it the Bayesian decision rule, which is also known as the minimum-error-probability (MEP) or maximum a posteriori probability (MAP) decision rule. What is the average error probability of the Bayesian decision rule? Is it smaller or larger than the average error probability of the maximum-likelihood decision rule? In the latter case, provide a brief explanation as to why the minimum-error-probability rule has a larger average error probability than another rule. Show that in each of the two cases i = 0 and for i = 1, it is true that if pi i > 0.8, then the Bayesian decision rule always decides that H, is the true hypothesis, no matter what the value of X is. Hint: Remember that pi 1-i = ? = 1 - pi i.Explanation / Answer
(a)
A
P(T+) = P(T+ | H+)P(H+) + P(T+ | H-)P(H-)
P(T+) = 0.999*0.02 + 0.01*(1-0.02) =0.02978
B
P(T+) = P(T+ | H+)P(H+) + P(T+ | H-)P(H-)
P(T+) = 0.99*0.02 + 0.001*(1-0.02) =0.02078
(b)
A
P(H+ | T+) = P(T+ | H+)P(H+)/P(T+) = 0.999*0.02/0.02978 =0.67092
B
P(H+ | T+) = P(T+ | H+)P(H+)/P(T+) = 0.99*0.02/0.02078 =0.95284
(c)
A
P(H+ | T-) = P(T- | H+)P(H+)/P(T-) = (1-0.999)*0.02/(1-0.02978) =0.00002061388
B
P(H+ | T-) = P(T- | H+)P(H+)/P(T-) = (1-0.99)*0.02/(1-0.02078) =0.00020424419