$8.$ In CBOW,

$1)[3$ pts$]$ Can we use a vocabulary index method $($instead of a one$-$hot vector$)$ as an input to CBOW

$2)[2$ pts$]$ The activation function of the output layer should be softmax. Explain the reason.

$3)[3$ pts$]$ After training is done in CBOW, how do we extract the word embedding vector for a certain word.

$9.[3$ pts$]$ In Skip$-$Gram, show that F$=$exp$($X$)$ satisfies the following:

F$(($w$\_$i$-$w$\_$j$)$w $\_$k $)=$F$($w$\_$i w $\_$k $)/$F$($w$\_$j w $\_$k $)$