$(26$ points$)$ Consider the Bayesian Network below:

Note: The numerical values of the probabilities are for part $($e$).$ You do not need to use them for

$($a$)-($d$).$

$($a$)$ Find the joint probability $P(A,B,C,D,E,F,G,H)$ as the product of conditional probabili$-$

ties, according to the graphical model given above.

$($b$)$ List all conditional independence given the graph.

$($c$)$ Show how to find the conditional probability $P(A|B).$

$($d$)$ Show how to find the marginal probability $P(A,E).$

$($e$)$ Using the conditional probability distribution $($CPD$)$ tables in the figure, find:

i$.P(a=0|b=1)$

ii$.P(a=0,e=1)$

iii. $P(a=0,b=0,c=1,d=1,e=1,f=0,g=0,h=0)$

iv$.P(b=0,c=1,d=1,f=0,g=0,h=0|a=0,e=1)$