Let the graph below depict a Restricted Boltzmann Machine with three hidden layer nodes labeled from left to right as $\text{'}a\text{'},\text{'}b\text{'}$ and $\text{'}c\text{'}$ and two visible layer nodes numbered $1$ and $2($from left to right$),$ none of which have any biases, with the following connecting weights:

$w_{1a}=2,w_{1b}=-1w_{1c}=1,w_{2a}=-2,w_{2b}=0w_{2c}=-1.$

Answer the following questions:

a$)$ If the initial visible vector $v=(1,1),$ what are the probabilities that the hidden layer nodes will be $1s?$

b$)$ Use the following random numbers to determine the states of the hidden layer nodes given the probabilities you obtained in part a$)$: $0\mathrm{.}87,0\mathrm{.}14,0\mathrm{.}64.$

c$)$ Given the states you obtained in part b$)$ for the hidden layer nodes, what are the probabilities that the visible layer nodes will be $1s?$

d$)$ Use the following random numbers to determine the reconstructed states of the visible layer nodes given the probabilities you obtained in part c$)$: $0\mathrm{.}25,0\mathrm{.}72.$