Consider a signal detection problem involving two hypotheses: H₁ : X = W and where H₁ : X = 8 + W, Ton || is a known signal vector, and W is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is P(H₁) = P(H₁) = 0.5. Suppose we observe X, but based on X we still determine that the two hypotheses are equally likely. What must be true about X?
Consider a signal detection problem involving two hypotheses: H₁ : X = W and where H₁ : X = 8 + W, Ton || is a known signal vector, and W is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is P(H₁) = P(H₁) = 0.5. Suppose we observe X, but based on X we still determine that the two hypotheses are equally likely. What must be true about X?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![Consider a signal detection problem involving two hypotheses:
\[
\mathcal{H}_0 : \vec{X} = \vec{W}
\]
and
\[
\mathcal{H}_1 : \vec{X} = \vec{s} + \vec{W},
\]
where
\[
\vec{s} =
\begin{bmatrix}
1 \\
2 \\
3 \\
4 \\
\end{bmatrix}
\]
is a known signal vector, and \(\vec{W}\) is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is \(P(\mathcal{H}_0) = P(\mathcal{H}_1) = 0.5\). Suppose we observe \(\vec{X}\), but based on \(\vec{X}\) we still determine that the two hypotheses are equally likely. What must be true about \(\vec{X}\)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc012e98d-11f3-4ebf-9e91-31a87ace6c91%2Fc79d42e7-b5f2-4384-a45e-d7fee20ab771%2Fqrj156g_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a signal detection problem involving two hypotheses:
\[
\mathcal{H}_0 : \vec{X} = \vec{W}
\]
and
\[
\mathcal{H}_1 : \vec{X} = \vec{s} + \vec{W},
\]
where
\[
\vec{s} =
\begin{bmatrix}
1 \\
2 \\
3 \\
4 \\
\end{bmatrix}
\]
is a known signal vector, and \(\vec{W}\) is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is \(P(\mathcal{H}_0) = P(\mathcal{H}_1) = 0.5\). Suppose we observe \(\vec{X}\), but based on \(\vec{X}\) we still determine that the two hypotheses are equally likely. What must be true about \(\vec{X}\)?
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