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Examination aids allowed: Open notes-books


• Read the questions carefully and answer only what is being asked.

• Answer all questions directly on the examination paper; use the last pages if you need more space, and provide clear pointers to your work.

• Show your intermediate work, and write clearly and legibly.

2. (20 points) Consider two random variables X and Y that are binary valued, i.e., the set of possible values is {0,1}. Let the marginal distribution of X be

It is given that the conditional distribution of Y given X is

(a) (10 points) Find the value of p ∈ [0,1] such that the marginal distributions of X and Y are the same.

(b) (10 points) For the value of p given in (a), find the conditional distribution of X given Y .

3. (20 points) Let (X,Y ) be a random vector whose distribution is given by

(a) (4 points) Find the marginal distributions of X and Y , and give their name and parameters.

(b) (16 points) Find the correlation coefficient Cor(X,Y ) = ρXY between X and Y .


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