GATE 2021 ST – Question 30
A Markov chain on {0,1} has transition rows (.25,.75),(.75,.25),and P(X₀=0)=P(X₀=1)=.5. Find Σ_(k=1)^100 E[(X_(2k))^(2k)].
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Correct answer: 50 to 50
Explanation
The initial distribution (.5,.5) is stationary,so P(X_j=1)=.5 for every j. A zero-one variable raised to any positive power equals itself. Each summand is therefore .5,and the sum is **50**.