Please help me with this questions. Please give detailed explanations. Thanks.

Let 0 = {HH, HT,TH, TT} be the sample space corresponding to two tosses of a coin. (For notational convenience, we're writing sample points as HH, instead of (H. H ) as we did in class.) Let X and Y be the number of heads on the first and second tosses respectively. That is X (HH) = X(HT) =1, X(TH) = X(TT) =0, Y(HH) = Y(TH) =1, Y(HT) = Y(TT) =0. (a) Enumerate o(X), and o(Y) explicitly. [Here (X) is the collection of all events that can be observed from X (ie. o (X) = {{X e A) |ACR) ). Similarly define o(Y).](b) Define a mass function p by P(HH) = P(HT) = P(TH) P(TT) = and let P denote the associated probability measure. Are X and Y indepen- dent under P? Justify your answer. (c) Define a second probability mass function p by 1 P( HH ) = P(HT) = P(TH) = 00 1 09 12: P(TT) = Do | 09 and let P be the associated probability measure. Are X and Y independent under P? Justify your