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Problem 3. Hypothesis test with a continuous observation 3 points possible (graded) Let $Theta$ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let $P(Theta = 1) = p$. Under the hypothesis $Theta = 0$, the random variable $X$ has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis $Theta = 1$, $X$ has a normal distribution with mean 2 and variance 1. Consider the MAP rule for deciding between the two hypotheses, given that $X = x$. 1. Suppose for this part of the problem that $p = 2/3$. The MAP rule can choose in favor of the hypothesis $Theta = 1$ if and only if $x ge c_1$. Find the value of $c_1$. $c_1 = $ 2. For this part, assume again that $p = 2/3$. Find the conditional probability of error for the MAP decision rule, given that the hypothesis $Theta = 0$ is true. $P(error|Theta = 0) = $ 3. 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.

          Problem 3. Hypothesis test with a continuous observation
3 points possible (graded)
Let $Theta$ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let $P(Theta = 1) = p$. Under the hypothesis $Theta = 0$,
the random variable $X$ has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis $Theta = 1$, $X$ has a normal
distribution with mean 2 and variance 1.
Consider the MAP rule for deciding between the two hypotheses, given that $X = x$.
1. Suppose for this part of the problem that $p = 2/3$. The MAP rule can choose in favor of the hypothesis $Theta = 1$ if and only if $x ge c_1$. Find
the value of $c_1$.
$c_1 = $
2. For this part, assume again that $p = 2/3$. Find the conditional probability of error for the MAP decision rule, given that the hypothesis
$Theta = 0$ is true.
$P(error|Theta = 0) = $
3. 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.

Problem 3. Hypothesis test with a continuous observation
3 points possible (graded)
Let Theta be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P(Theta = 1) = p. Under the hypothesis Theta = 0,
the random variable X has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis Theta = 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.
1. Suppose for this part of the problem that p = 2/3. The MAP rule can choose in favor of the hypothesis Theta = 1 if and only if x ge c1. Find
the value of c1.
c1 =
2. For this part, assume again that p = 2/3. Find the conditional probability of error for the MAP decision rule, given that the hypothesis
Theta = 0 is true.
P(error|Theta = 0) =
3. 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.

Added by Christopher J.

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