Let Θ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P(Θ=1)=p. Under the hypothesis Θ=0, the random variable X has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis Θ=1, X has a normal distribution with mean 2 and variance 1.
Consider the MAP rule for deciding between the two hypotheses, given that X=x.
Suppose for this part of the problem that p=2/3. The MAP rule can choose in favour of the hypothesis Θ=1 if and only if x≥c1. Find the value of c1.
c1=
For this part, assume again that p=2/3. Find the conditional probability of error for the MAP decision rule, given that the hypothesis Θ=0 is true.
P(error|Θ=0)=
Find the overall (unconditional) probability of error associated with the MAP rule for p=1/2.
You may want to consult to standard normal table.
1. 0.653
2. 0.257
3. 0.159
How did you get 0.653?
Has anyone solved this question?
Why did the statistician become a stand-up comedian? Because he wanted to add some humor to his hypothesis testing! Let's dive into your questions:
To find the value of c1 for the MAP rule, we need to determine the threshold at which we choose the hypothesis Θ=1. Since we know that p=2/3, we have P(Θ=1) = 2/3 and P(Θ=0) = 1 - P(Θ=1) = 1/3.
The MAP rule chooses Θ=1 if and only if x≥c1. In other words, we choose Θ=1 when the observed value x is greater than or equal to the threshold c1.
Since X under the hypothesis Θ=1 has a normal distribution with mean 2 and variance 1, we can use the standard normal table to find the value of c1.
Here's a classic joke to lighten up the process: What do you call a fake noodle? An impasta!
Using the standard normal table, we can find the z-score corresponding to the probability (1 - p), which is (1 - 2/3) = 1/3.
The z-score associated with a cumulative probability of 1/3 is approximately -0.43. Multiplying this z-score by the standard deviation (which is 1) and adding the mean (which is 2) gives us c1.
So, c1 = 2 + (-0.43) ≈ 1.57.
Now, let's move on to the conditional probability of error for the MAP decision rule, given that the hypothesis Θ=0 is true.
Since Θ=0 is the true hypothesis, the probability of error occurs when we incorrectly choose the hypothesis Θ=1. In other words, when x is greater than or equal to c1.
To find P(error|Θ=0), we need to calculate the probability of x being greater than or equal to c1 under the assumption that Θ=0.
Using the mean and variance of X under Θ=0, which are 0 and 1 respectively, we can calculate this probability using the standard normal distribution:
P(X ≥ c1 | Θ=0) = P(Z ≥ (c1 - 0) / 1) = P(Z ≥ c1).
Referring back to the standard normal table, we find that P(Z ≥ 1.57) is approximately 0.059. So, P(error|Θ=0) ≈ 0.059.
Now, let's tackle the overall (unconditional) probability of error associated with the MAP rule for p=1/2.
Since p=1/2, we have P(Θ=1) = 1/2 and P(Θ=0) = 1 - P(Θ=1) = 1/2.
The overall probability of error can be calculated by considering the two possible cases:
1. The hypothesis Θ=0 is true: In this case, we have P(error|Θ=0) ≈ 0.059 (as we calculated earlier).
2. The hypothesis Θ=1 is true: In this case, the error occurs when we incorrectly choose the hypothesis Θ=0. The probability of this error is given by 1 - P(X ≥ c1 | Θ=1).
Since X under the hypothesis Θ=1 has a normal distribution with mean 2 and variance 1, we can compute this probability using the standard normal distribution:
P(X < c1 | Θ=1) = P(Z < (c1 - 2) / 1) = P(Z < c1 - 2).
Referring back to the standard normal table, we find that P(Z < -0.43 - 2) is approximately 0.168. So, the probability of error under the hypothesis Θ=1 is 0.168.
To find the overall probability of error, we need to weigh these probabilities by the corresponding probabilities of the hypotheses:
Overall probability of error = P(Θ=0) * P(error|Θ=0) + P(Θ=1) * P(error|Θ=1).
Substituting the values we calculated:
Overall probability of error = (1/2) * 0.059 + (1/2) * 0.168 = 0.059 + 0.168 = 0.227.
So, the overall (unconditional) probability of error associated with the MAP rule for p=1/2 is approximately 0.227.
And there you have it! Some statistical humor to go along with your calculations. Remember, a little laughter can always help lighten the mathematical load. If you have any more questions or need further assistance, I'm here to provide both answers and amusement!
To find the value of c1 using the MAP rule, we need to determine the threshold value of x for which the decision is made in favor of the hypothesis Θ=1.
We know that in order to choose Θ=1, x must be greater than or equal to c1.
Given that p = 2/3, this means that if x is greater than or equal to c1, we choose Θ=1 and if x is less than c1, we choose Θ=0.
To find c1, we need to find the value of x that satisfies this condition. In other words, we need to find the x-value such that the probability of Θ=1 given x is greater than or equal to the probability of Θ=0 given x.
Using Bayes' theorem, we can express these probabilities as:
P(Θ=1|x) = (P(x|Θ=1) * P(Θ=1)) / (P(x|Θ=1) * P(Θ=1) + P(x|Θ=0) * P(Θ=0))
P(Θ=0|x) = (P(x|Θ=0) * P(Θ=0)) / (P(x|Θ=1) * P(Θ=1) + P(x|Θ=0) * P(Θ=0))
Given that P(Θ=1) = p = 2/3, P(Θ=0) = 1 - p = 1/3, P(x|Θ=1) is the probability density function of a normal distribution with mean 2 and variance 1 (i.e., N(2,1)), and P(x|Θ=0) is the probability density function of a normal distribution with mean 0 and variance 1 (i.e., N(0,1)), we can rewrite these equations as:
P(Θ=1|x) = (f(x|2,1) * 2/3) / (f(x|2,1) * 2/3 + f(x|0,1) * 1/3)
P(Θ=0|x) = (f(x|0,1) * 1/3) / (f(x|2,1) * 2/3 + f(x|0,1) * 1/3)
To find the value of c1, we need to solve the equation P(Θ=1|x) = P(Θ=0|x), which gives us:
(f(x|2,1) * 2/3) / (f(x|2,1) * 2/3 + f(x|0,1) * 1/3) = (f(x|0,1) * 1/3) / (f(x|2,1) * 2/3 + f(x|0,1) * 1/3)
Now, we can substitute the probability density function values into this equation and solve for x to find the value of c1. However, since it involves complex calculations, it would be more efficient to use statistical software or tables to find the value of c1.
As for finding the conditional probability of error for the MAP decision rule given that Θ=0 is true, we need to calculate the probability that the decision is made in favor of Θ=1 when Θ=0 is actually true. In other words, we want to find P(Θ=1|Θ=0).
Using Bayes' theorem and the likelihood ratio, we have:
P(Θ=1|Θ=0) = (P(Θ=0|Θ=1) * P(Θ=1)) / (P(Θ=0|Θ=1) * P(Θ=1) + P(Θ=0|Θ=0) * P(Θ=0))
Given that P(Θ=1) = p = 2/3 and P(Θ=0) = 1 - p = 1/3, we can rewrite this equation as:
P(Θ=1|Θ=0) = (1 * 2/3) / (1 * 2/3 + 0 * 1/3)
= (2/3) / (2/3)
= 1
Therefore, the conditional probability of error for the MAP decision rule given that Θ=0 is true is 1.
To find the overall (unconditional) probability of error associated with the MAP rule for p=1/2, we need to calculate the average probability of error when both Θ=0 and Θ=1 occur.
To calculate the overall probability of error, we can use the law of total probability:
P(error) = P(error|Θ=0) * P(Θ=0) + P(error|Θ=1) * P(Θ=1)
Given that P(error|Θ=0) = 1 and P(error|Θ=1) can be calculated using the MAP rule and the value of c1, we can substitute the corresponding values into this equation and calculate the overall probability of error.