Q 5.30 An image is partitioned into two... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 5: Q 5.30 (page 214)

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with $渭 = 4$ and $蟽^{2} = 4$ , whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters $(6, 9)$ . A point is randomly chosen on the image and has a reading of $5$ . If the fraction of the image that is black is $伪$ , for what value of $伪$ would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?

Short Answer

Expert verified

Therefore, the value of $伪 = 0.3827$ $伪 = 0.3827$ .

Step by step solution

Given information:

A reading taken from a randomly chosen point in the white section will be normally distributed with $渭 = 4$ and $蟽^{2} = 4$ , whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters $(6, 9)$

Explanation:

$P (g i v e n p o i n t b e l o n g t o b l a c k r e g i o n) = \frac{1}{2}$

Generally,

Equation 1:

$P (g i v e n p o i n t b e l o n g t o b l a c k r e g i o n) = \frac{伪脳 P (v a l u e = \frac{5}{b l a c k})}{伪脳 P (v a l u e = \frac{5}{b l a c k}) + (1 - 伪) 脳 P (v a l u e = \frac{5}{w h i t e})}$

Explanation:

Let's put them equal to each other:

Put equation 1 $\frac{伪脳 \frac{e^{\frac{- 1}{8}}}{2 \sqrt{(2 脳蟺)}}}{伪脳 \frac{e^{\frac{- 1}{8}}}{2 \sqrt{(2 脳蟺)}} + (1 - 伪) \frac{e^{\frac{- 1}{8}}}{3 \sqrt{(2 脳蟺)}}} = \frac{1}{2} \frac{\frac{伪}{2}}{\frac{伪}{2} + \frac{(1 - 伪) 脳 e^{\frac{- 5}{72}}}{3}} = \frac{1}{2} \frac{3 伪}{3 伪 + 2 (1 - 伪) 脳 e^{\frac{- 5}{72}}} = \frac{1}{2} 6 伪 = 3 伪 + 2 (1 - 伪) 脳 e^{\frac{- 5}{72}} 3 伪 = 1.86 - 1.86 伪伪 = 0.3827$

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