Problem 6 (15 pts) Let X, Y be two random variables with a joint distribution P. There are five possible values of X: X1

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Problem 6 15 Pts Let X Y Be Two Random Variables With A Joint Distribution P There Are Five Possible Values Of X X1 1 (98.64 KiB) Viewed 84 times

Problem 6 (15 pts) Let X, Y be two random variables with a joint distribution P. There are five possible values of X: X1, X2, X3,X4, X5, and five possible values of Y: Y1, ...,Y5. Consider the 5 x 5 matrix M such that Mij = P(Y = y;|X = xi), = = i = 1,. 5, j = 1, ...,5. (A) Suppose X, Y are independent. Show that rank(M) = 1. (B) Show that the vector (1,1, 1, 1, 1)T is an eigenvector of M. What is the corresponding eigenvalue of M? (C) Let there be an additional random variable Z, taking 2 possible values: Z1,...,z2. Suppose that X, Y, Z satisfy the following" P(X, Z|Y) = P(Z|Y)P(X|Z). = Show that rank (M) < 2.