For the Transition matrix P= [ 0.5 0.5 ], solve the equation SP=S to find the stationary matrix S and...

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x = xxxx xxxx
xxxx xxxx

xxx S xx xx xx
xxxx xxxx equation SP=S
xx get,

[x y] xxxx xxxx = [x xx
xxxx 0.4]

xxx [(0.5x x xxxxx xxxxx + xxxxxx = [x xx
by xxxxxxxxx xxxx xxxxxxx xx xxxx
0.5x + xxxx x x xxx xxxx x 0.4y x x

xx > 0.6y x xxxx and xxxx = xxx

and xx xxxx xxxx x + y x x
xxxxxx xxxx x x xxx
xxx xxxx xxxx x xxxx
xx xxxx 0.6y x xxxxx x y)
==> xxxx = xxx
==> x x xxxxxxx x xxxx x xxxxx
xxx xxxxxx x = 1-y = xxxxxx = xxxxx

hence, xxxxxxxxxx xxxxxx S = xxxxxx xxxxxx

xxxxxx x is aperiodic {since diagonal entries are non xxxxx and irreducible {since, there xxx xxxx xxx states xxx prob. xx xxxxx from xxx state xx another is non zero}

xxxxxx for xxxxxxxxxxx xxx xxxxxxxxx xxxxxx limiting xxxxxx is xxxxx by xxxxxx xxxx xxxxxxxxxx xxxxx xx each xxxx
xxxxxx limiting xxxxxx xxx this P is given xx
x x [0.545 xxxxxx
xxxxxx 0.455]