Hello, I have questions about Probabilistic Graphical Models. I would like to make some ways but I didn't answer questions because I don't understand this topic. Could you please help me with these questions?

3. Properties of independent variables. Assume that A and B are independent random variables. Which of the following options are always true? You may select 1 or more options. P(B|A) = P(B) P(A B) = P(A) P(A) + P(B) = 1 P(A) = P(B) 4. Factor marginalization. Let X", Z be binary variables, and let Y" be a variable that takes on values 1, 2, or 3. If d(X, Y. Z) is the factor shown below, compute the entries of the factor "(Y, Z) = L. (X, Y,Z). giving your answer according to the ordering of assignments to variables as shown below. Separate the 4 entries of the factor with spaces, and do not add any extra trailing or leading zeroes or decimal points. x Y Z .(X, Y,Z) 1 1 1 1 1 2 95 1 2 1 65 Y Z y(Y.Z) 12 2 1 3 1 57 12 13 2 S 2 1 2 1 1 40 2 2 212 40 3 1 2 2 1 14 132 2 2 2 7H 18 29 2 Enter answer here1. Factor product. 1 poirn Let X, Y and Z be binary variables. If d. (X, Y) and da (Y', 2) are the factors shown below, compute the selected entries (marked by a ?) in the factor "(X, Y. 2) =(X, Y). (Y, Z). giving your answer according to the ordering of assignments to variables as shown below. Separate each of the 3 entries of the factor with spaces, e.g., an answer of 0.1 0.2 0.3 means that w( 1, 1, 1) = 0.1, w(1, 2,1) = 0.2. and (2, 2, 2) = 0.3. Give your answers as exact decimals without any trailing zeroes. XY Z (X,F.Z] 1 1 1 xroux. r 1 1 2 1 1 1 1 02 12 1 1 2 0.5 1 2 0.2 12 2 2 1 OG 2 1 2 1 1 2 2 0.6 2 2 1.0 2 1 2 2 2 1 2 2 2 Enter answer here 2. Factor reduction. 1 point Let X", Z be binary variables, and let Y" be a variable that takes on values 1, 2, or 3. Now say we observe Y" = 3. If @(_X, Y, 2) is the factor shown below, compute the missing entries of the reduced factor 4(X, Z) given that Y" = 3, giving your answer according to the ordering of assignments to variables as shown below. As before you may separate the 4 entries of the factor by spaces XY Z(X, Y.Z) 1 1 1 14 1 12 60 1 2 1 40 1 2 2 27 X ZW( X.Z 1 3 1 42 7 3 2 85 12 21 1 2 1 2 1 2 59 2 2 2 2 1 54 2 2 2 96 2 E 30 Enter answer here