Question

Suppose X is a random variable that, under hypothesis $H_0$, has PDF egin{equation} f_X(x mid H_0) = egin{cases} (2/3)(x+1), & 0 le x le 1\ 0, & ext{otherwise} end{cases} end{equation} and, under hypothesis $H_1$, has PDF egin{equation} f_X(x mid H_1) = egin{cases} 1, & 0 le x le 1\ 0, & ext{otherwise} end{cases} end{equation} (a) Find the Bayes rule and minimum Bayes risk for testing $H_0$ versus $H_1$ with uniform costs (i.e., $C_{00} = C_{11} = 0$ and $C_{10} = C_{01} = 1$) and equal priors (i.e., $p_0 = p_1 = 1/2)$. (b) Find the Neyman-Pearson rule for false-alarm probability $alpha in (0, 1)$. Calculate the corresponding detection probability.

          Suppose X is a random variable that, under hypothesis $H_0$, has PDF
egin{equation*}
f_X(x mid H_0) = egin{cases} (2/3)(x+1), & 0 le x le 1\ 0, & 	ext{otherwise} end{cases}
end{equation*}
and, under hypothesis $H_1$, has PDF
egin{equation*}
f_X(x mid H_1) = egin{cases} 1, & 0 le x le 1\ 0, & 	ext{otherwise} end{cases}
end{equation*}
(a) Find the Bayes rule and minimum Bayes risk for testing $H_0$ versus $H_1$ with uniform costs
(i.e., $C_{00} = C_{11} = 0$ and $C_{10} = C_{01} = 1$) and equal priors (i.e., $p_0 = p_1 = 1/2)$.
(b) Find the Neyman-Pearson rule for false-alarm probability $alpha in (0, 1)$. Calculate the corresponding detection probability.

Suppose X is a random variable that, under hypothesis H0, has PDF
eginequation*
fX(x mid H0) = egincases (2/3)(x+1), 0 le x le 1 0, extotherwise endcases
endequation*
and, under hypothesis H1, has PDF
eginequation*
fX(x mid H1) = egincases 1, 0 le x le 1 0, extotherwise endcases
endequation*
(a) Find the Bayes rule and minimum Bayes risk for testing H0 versus H1 with uniform costs
(i.e., C00 = C11 = 0 and C10 = C01 = 1) and equal priors (i.e., p0 = p1 = 1/2).
(b) Find the Neyman-Pearson rule for false-alarm probability alpha in (0, 1). Calculate the corresponding detection probability.

Added by Catherine D.

Question

Please give Ace some feedback