Consider there are two populations one with attribute D = 1 and the other with attribute D = 0. A binary classifier with threshold a is used to categorize a given sample into one of the two populations. The classifier returns the result T = 1 if it determines the sample to be of type D = 1 otherwise it returns T = 0 and determines that the sample is of type D = 0. The random variable X measured by the classifier is distributed as follows: X|(D = 1) ~ Unif(10, 30) and X|(D = 0) ~ Unif(0, 20). 1. What is the Sensitivity (True Positive Rate)? 2. What is the Specificity (True Negative Rate)?
Consider there are two populations one with attribute D = 1 and the other with attribute D = 0. A binary classifier with threshold a is used to categorize a given sample into one of the two populations. The classifier returns the result T = 1 if it determines the sample to be of type D = 1 otherwise it returns T = 0 and determines that the sample is of type D = 0. The random variable X measured by the classifier is distributed as follows: X|(D = 1) ~ Unif(10, 30) and X|(D = 0) ~ Unif(0, 20). 1. What is the Sensitivity (True Positive Rate)? 2. What is the Specificity (True Negative Rate)?
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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
Transcribed Image Text:4 Problem
Consider there are two populations one with attribute D = 1 and the other with attribute D = 0. A binary
classifier with threshold a is used to categorize a given sample into one of the two populations. The classifier
returns the result T = 1 if it determines the sample to be of type D = 1 otherwise it returns T = 0 and
determines that the sample is of type D = 0. The random variable X measured by the classifier is distributed
as follows: X|(D = 1) ~ Unif(10, 30) and X|(D = 0) ~ Unif(0, 20).
1. What is the Sensitivity (True Positive Rate)?
2. What is the Specificity (True Negative Rate)?
Expert Solution
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Question 1
Sensitivity (True Positive Rate) is the probability that the classifier correctly identifies a sample as belonging to the population with attribute D = 1, given that the sample actually belongs to that population. Mathematically, it can be expressed as:
Sensitivity = P(T = 1 | D = 1)
Since X | (D = 1) ~ Unif(10, 30), the probability that X falls in the range [a, 30] given that D = 1 is:
P(X >= a | D = 1) = (30 - a) / 20
Therefore, the sensitivity is:
Sensitivity = P(T = 1 | D = 1) = P(X >= a | D = 1) = (30 - a) / 20
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