Please see the question in the attached. Thanks.

Consider a random experiment where 3 fair coins are tossed and the outcomes are recorded. Let 9 denote the sample space. Note that [9| 2 23. Let .7: = 29, the power set. Let X denote the number of HEADS observed. Note that X maps (2 to 9X, where 52X 2 {0,1,2,3}. Let J-"X : 29X, the power set. Note that the function X is measurable with respect to (9);)a(QXan)' 0 Find the smallest aalgebra if that is a subset of J: such that the function X is measur able with respect to (9,55), (9X,.7-'X). 0 Let Y denote the number of runs of TAILS observed. For example Y(H H H ) = O, Y(TTH) = 1, Y(THT) = 2 and so on. Note that fly 2 {0,1,2}. Note that neither X is a function of Y, nor Y is a function of X. Let FY = 295/, the power set. Is Y measurable with respect to (Q, .73), (Qy, fy)