In Euclidean spaces, a set C is called a convex set if for all x and y in C, the line segment connecting x and y is included in C. In other words, the affine combination tx p1 tqy belongs to C, for all x, y P C, and t P r0, 1s. Prove that if a Markov Chain has stationary distribution, then all stationary distribution of this Markov Chain is a convex set. (b) Let pXnqn0,1,2 be a Markov chain with state space S t1, 2, , Nu. The sum of each column of the stochastic probability matrix P is 1, i. e. ¸ N i1 Pij 1, for all j Show that if this Markov chain is aperiodic and irreducible, then the stationary distribut

The null and alternate hypotheses are: h0: μ1 ≤ μ2 h1: μ1 > μ2 a random sample of 22 items from the first population showed a mean of 113 and a standard deviation of 12. a sample of 16 items for the second population showed a mean of 99 and a standard deviation of 6. use the 0.01 significant level. find the degrees of freedom for unequal variance test. (round down your answer to the nearest whole number.) state the decision rule for 0.010 significance level. (round your answer to 3 decimal places.) compute the value of the test statistic. (round your answer to 3 decimal places.) what is your decision regarding the null hypothesis? use the 0.01 significance level.