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I'm at the tail end of a bioinformatics class and our exam is on Monday. I think one of our homework problems was quite interesting and fun.

Q1.
Prove that the stationary distribution of any symmetric Markov chain is the discrete uniform distribution.

Q2.
Prove that the stationary distribution of any Markov chain with the following properties is the discrete uniform distribution.
Properties:
The sum of M(i,j) over i =1 for all i. Similarly for j.

Obviously if you prove Q2, you don't need to prove Q1. You might want to include a sentence on how Q1 is within Q2.

Ok, so we start form here I typed in in LaTeX and got it on an image board



And if they're all the same, then they're equal. Then pi_i forms the uniform distribution.