For each of the following transition matrices, determine whether the Markov chain with that transition matrix is regular: (1) Is the Markov chain whose transition matrix whose transition matrix is 0 0.5 0.5 0.5 0 0.5 0 0 regular? (Yes or No) (2) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0 0.7 0 0 regular? (Yes or No) (3) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.6 0 0.4 1 0 0 regular? (Yes or No) (4) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0 0.6 0 0.4 regular? (Yes or No) (5) Is the Markov chain whose transition matrix whose transition matrix is 0 1 0 0.3 0.2 0.5 0 1 0Let X and Y be independent /(0, 1) distributed random variables. Show that X + Y and X - Y are independent N(0, 2) distributed random variables Let X and Y be independent Exp(1)-distributed random variables. Show that X/(X + Y) and X + Y are independent, and find their distributions. Show that if X E N(0, 1) and Y E x2(n) are independent random variables then X/ VYet(n). The random variables X1 and X2 are independent and equi-distributed with density function 4x3, for 0