Decision making

The folowing concepts mathematically and give some examples (numerically) which are necessary in probability theory event sample sponce - statistical inference descriptive s. permutasyou kon binasyon - clensity function - conditional probability - multiplication rule - dependent event - Independent event Boyes Rule . P.s ; The find and discuit the validated of Bayes' Rule .0.4 points " Which Chi Square distribution looks the most like a normal distribution? a A Chi Square distribution with 4 degrees of freedom b) A Chi Square distribution with 5 degrees of freedom CA Chl Square distribution, with 6 degrees of freedom of A Chi Square distribution with 16 degrees of freedom1. (a) Explain what is meant by the transition probability matrix of a homogeneous Markov chain. [5 marks] (b) Explain what is meant by the stationary distribution of a Markov chain? [5 marks] (c) A Markov chain has transition probability matrix, A, with entries ay, and stationary distribution . Write down an expression for the entries of the reverse Markov chain. [5 marks (d) Consider the following transition probability matrix of a homogo- neous Markov chain, with three states i,j and & (the TPM is in that order). If the stationary vector of the chain is (1/9, 2/9, 2/3), determine whether the Markov chain is reversible. 1 Ik 1- /0.2 0.2 0.6 0.1 0.6 0.3 K 0.1 0.1 0.8 [5 marks] (e) Let X1, X2, X, be a sequence of random variables resulting from the above Markov chain. If X = i and Xy = j what is the probability that X2 = k? [5 marks]State-space equation of a linear autonomous system are given below -5/3 4/3 a x ( t ) = - 1/ 3 0 7/3 O - 2 a ) find the eat state transition matrix of system by the Lapalace transform b ) find the At state transition matrix of system with the Cayley - Hamilton Theorem ( ) find the efit state transition matrix of system using eigenvectors