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represented by a scalar x with Gaussian class-conditional probability density functions (pdf) p(x | wi)~ N(14, 07) (1 ≤i≤3). Specifically, we have 141=-1, 142=0, s= 2 and 1=2, 021, 03 = 3. Please answer the following questions: a) In case of minimum-error-rate classification, which state of nature should we decide for x-1.5? b) Suppose we have a set of two possible actions A={a₁, 2) with the following loss function: Action Class α = α1 x=82 W=11 5 10 w = wa 10 30 W=W3 20 3 For an observation x with value 0.5, which action should we apply on x?

represented by a scalar x with Gaussian class-conditional probability density functions (pdf) p(x | wi)~ N(14, 07) (1 ≤i≤3). Specifically, we have 141=-1, 142=0, s= 2 and 1=2, 021, 03 = 3. Please answer the following questions: a) In case of minimum-error-rate classification, which state of nature should we decide for x-1.5? b) Suppose we have a set of two possible actions A={a₁, 2) with the following loss function: Action Class α = α1 x=82 W=11 5 10 w = wa 10 30 W=W3 20 3 For an observation x with value 0.5, which action should we apply on x?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Given a set of three states of nature \( \Omega = \{\omega_1, \omega_2, \omega_3\} \) with prior probabilities \( P(\omega_1) = 0.2 \), \( P(\omega_2) = 0.3 \), and \( P(\omega_3) = 0.5 \). In addition, each observation is
Transcribed Image Text:Given a set of three states of nature \( \Omega = \{\omega_1, \omega_2, \omega_3\} \) with prior probabilities \( P(\omega_1) = 0.2 \), \( P(\omega_2) = 0.3 \), and \( P(\omega_3) = 0.5 \). In addition, each observation is
The image contains a problem related to Gaussian class-conditional probability density functions and decision-making in the context of machine learning or statistical classification. Here is the transcription and description as if it were for an educational website: --- In the context of a decision-making problem, we consider a scalar \( x \) with Gaussian class-conditional probability density functions (pdf) expressed as \( p(x \mid \omega_i) \sim \mathcal{N}(\mu_i, \sigma_i^2) \) for \( 1 \leq i \leq 3 \). Specifically, the parameters of the distributions are given as: \[ \mu_1 = -1, \quad \mu_2 = 0, \quad \mu_3 = 2 \] \[ \sigma_1 = 2, \quad \sigma_2 = 1, \quad \sigma_3 = 3 \] Please address the following questions: a) In the case of minimum-error-rate classification, which state of nature should we decide for \( x = 1.5 \)? b) Suppose we have a set of two possible actions \( A = \{\alpha_1, \alpha_2\} \) with the following loss function: | Class | \(\omega = \omega_1\) | \(\omega = \omega_2\) | \(\omega = \omega_3\) | |-------|------------------|------------------|------------------| | \(\alpha = \alpha_1\) | 5 | 10 | 20 | | \(\alpha = \alpha_2\) | 10 | 30 | 3 | For an observation \( x \) with value 0.5, which action should we apply on \( x \)? --- This problem illustrates decision theory concepts using Gaussian distributions to model the likelihood of different classes and a loss matrix to evaluate decision-making actions based on observed data.
Transcribed Image Text:The image contains a problem related to Gaussian class-conditional probability density functions and decision-making in the context of machine learning or statistical classification. Here is the transcription and description as if it were for an educational website: --- In the context of a decision-making problem, we consider a scalar \( x \) with Gaussian class-conditional probability density functions (pdf) expressed as \( p(x \mid \omega_i) \sim \mathcal{N}(\mu_i, \sigma_i^2) \) for \( 1 \leq i \leq 3 \). Specifically, the parameters of the distributions are given as: \[ \mu_1 = -1, \quad \mu_2 = 0, \quad \mu_3 = 2 \] \[ \sigma_1 = 2, \quad \sigma_2 = 1, \quad \sigma_3 = 3 \] Please address the following questions: a) In the case of minimum-error-rate classification, which state of nature should we decide for \( x = 1.5 \)? b) Suppose we have a set of two possible actions \( A = \{\alpha_1, \alpha_2\} \) with the following loss function: | Class | \(\omega = \omega_1\) | \(\omega = \omega_2\) | \(\omega = \omega_3\) | |-------|------------------|------------------|------------------| | \(\alpha = \alpha_1\) | 5 | 10 | 20 | | \(\alpha = \alpha_2\) | 10 | 30 | 3 | For an observation \( x \) with value 0.5, which action should we apply on \( x \)? --- This problem illustrates decision theory concepts using Gaussian distributions to model the likelihood of different classes and a loss matrix to evaluate decision-making actions based on observed data.
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