Caution
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Bayesian Statistics: Mixture Models
Mixture Models, Homework
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---
title : 'Homework The Likelihood function - M1L2HW1'
subtitle : 'Bayesian Statistics: Mixture Models'
categories:
- Bayesian Statistics
keywords:
- Mixture Models
- Homework
---
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Consider a random sample $(3.5,9.7,8.2,6.4,7.1)$ composed of $n=5$ observations form the mixture with density:
$$
f(x)=w\lambda_1 e^{-\lambda_1 x} + (1-w)\lambda_2 e^{-\lambda_2 x}
$$
What is the complete-data likelihood associated with the indicator vector
$(1,1,2,1,2)$
- [ ] $w^{(1-w)^4} \lambda_1 e^{-22.7 \lambda_1} \lambda_2^4 e^{-11.3\lambda_2}$
- [ ] $w^3(1-w)^2 \lambda_1^2 e^{-15.3\lambda_1} \lambda_2^3 e^{-19.6\lambda_2}$
- [x] $w^3(1-w)^2 \lambda_1^3 e^{-19.6\lambda_1} \lambda_2^2 e^{-15.3\lambda_2}$
2. Consider a random sample $(3.5,9.7,7.1)$ composed of $n=3$ observations form the mixture with density:
$$
f(x)=w\lambda_1 e^{-\lambda_1 x} + (1-w)\lambda_2 e^{-\lambda_2 x}
$$
The observed-data likelihood $L_{O}(w,\lambda_1,\lambda_2;x)$ for this sample is given by the product
$$
\begin{aligned}
& \left\{ w\lambda_1 e^{-\lambda_1 3.5} + (1-w)\lambda_2 e^{-\lambda_2 3.5} \right\} \\
\times &\left\{ w\lambda_1 e^{-\lambda_1 9.7} + (1-w)\lambda_2 e^{-\lambda_2 9.7} \right\} \\
\times &\left\{ w\lambda_1 e^{-\lambda_1 7.1} + (1-w)\lambda_2 e^{-\lambda_2 7.1} \right\}
\end{aligned}
$$
- [x] yes
- [ ] no
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