We want to create a generative binary classification model for classifying non$-$negative onedimensional data. This means, that the labels are binary $($y in $\{0,1\})$ and the samples are

x in $[0,\backslash$infty $).$ We assume uniform class probabilities

p$($y $=0)=$ p$($y $=1)=1$

$2$

$1$

SSY$316-$ Homework $3$

As our samples x are non$-$negative, we use exponential distributions $($and not Gaussians$)$ as

class conditionals:

p$($x $|$ y $=0)=$ Expo$($x $|\backslash$lambda $0)$ and p$($x $|$ y $=1)=$ Expo$($x $|\backslash$lambda $1)$

where $\backslash$lambda $06=\backslash$lambda $1.$ Assume, that the parameters $\backslash$lambda $0$ and $\backslash$lambda $1$ are known and fixed.

$1.$ What is the name of the posterior distribution p$($y $|$ x$)?$ You only need to provide the

name of the distribution $($e$.$g$.,$normal$,$gamma$,$ etc.$),$ not estimate its parameters.

$2.$ What values of xare classified as class $1?($As usual, we assume that the classification

decision is y$=$ argmaxkp$($y $=$ k $|$ x$))$