(0 1 0 5. (9pts) Let P=1 0 11 (a) Calculate PPT 0 0 0 1 -1 0 0 0 -1 0 0 0

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mdasfaaruddinshareefLv10

15 Jun 2023

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Nelly StrackeLv2

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10. Which of the following is a one-dimensional subspace of R3? A) {(2, y, z)?|x2 – y + 2z=0}. B) {(x,y, 2)7|x + 24 +32 =0}. C) The solution set of Av = 0 where A= 1 -2 3 -1 4 0 0 2 3 D) {(x, y, z)?|x + 2y + 32 = 4}. E) The solution set of Av = 0 where A= 1 0 2 0 2 0 11 0 5

3. Consider any Random Walk W = (Wn)n≥0 on Z, that is, Wn := a + X1 + · · · + Xn whereX1,X2,... areIIDRVs.
Let r := P0(return to 0) be the probability that W returns to 0 given it starts at 0. Let N be the total number of visits to 0 including the visit at time 0. Since intuitively the Random Walk ‘starts afresh’ whenever it first returns to 0, show that for r ∈ [0, 1),

P0(N =k)=rk−1(1−r) (k=1,2,...),

so that the total number of visits to 0 when started at 0 is a Geometrically dis- tributed RV. Also note that P0(N = +∞) = 1 when r = 1.
Hence, calculate the expected total number of visits to 0 when starting at 0, E0(N), in terms of r.

On the other hand, with ξn := I{Wn=0}, note N = ξ0 +ξ1 +ξ2 +... where the sum counts 1 for every time 0 is visited. By taking expectations and comparing expressions, deduce that

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