Equation(1):
0), and Vi(n) i rst-step analysis to show that
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2. Consider a two state DTMC with state space E = {1 ,2). Let T = min(n > 0 : Xn-1) (i) Compute E...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1).
Consider a...
5. Let (Xn)n be a Markov chain on a state space S with n-step transition probabilities...
5. Let (Xn)n be a Markov chain on a state space S with n-step transition probabilities PTy = P(X,= y|Xo = x). Define (n) N x Xn=r n0 and U(G,) ΣΡ. n0 Show that (a) U(x, y)ENy|Xo= x] and (b) U(a, y) P(T, < +o0|X0= x)U(y, y), where Ty = inf {n 2 0 : X y}.
Let X0,X1,... be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1 = in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following...
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose...
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that T has a geometric distribution with respect to the conditional probability P
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that...
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that...
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that for any n, m-1 and JAn+m, . . . , İn+1,in-1 , . . . , io є S.
4. Fix > 0. For n > λ let Xn be Geometric(A/n). Show that X n/n...
4. Fix > 0. For n > λ let Xn be Geometric(A/n). Show that X n/n converges in distribution to an Exponential(A). (Hint: again, compute moment generating functions.)
Q5. Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, ·...
Q5. Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · } and transition matrix (pij ). Find (in terms QA for appropriate A) P{ max 0≤k≤n Xk ≤ m|X0 = i} . Q6. (Flexible Manufacturing System). Consider a machine which can produce three types of parts. Let Xn denote the state of the machine in the nth time period [n, n + 1) which takes values in {0, 1, 2, 3}. Here...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) =...
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) = E (eitX»), ER be their characteristic functions. Let Yn = {k=0 Xk, n > 0, X0 = 0, and 8. () = {1*: (),ER. k = 1 a) Let t be so that I1=1 løk (t)) > 0. Show that _exp{itYn} ?, n > 0, On (t) is a martingale with respect to Fn = (Xo, ...,Xn), n > 0, and sup, E (M,|2)...
Consider a DTMC X;n 2 0 with state space E 0,1,2,... ,N), and transition probability matrix P = (pij). Define T = min(n > 0 : Xn-0), and vi(n) = P(T > n|X0 = i). Use the first-step analysis to show that vi (72), t"2(n), . . . , UN(n)) = where B is a submatrix of P obtained by deleting the row and column corresponding to the state 0. Hint: First establish a recursive formula v(n )-ΣΝ1pijuj(n-1).
Consider a...
5. Let (Xn)n be a Markov chain on a state space S with n-step transition probabilities PTy = P(X,= y|Xo = x). Define (n) N x Xn=r n0 and U(G,) ΣΡ. n0 Show that (a) U(x, y)ENy|Xo= x] and (b) U(a, y) P(T, < +o0|X0= x)U(y, y), where Ty = inf {n 2 0 : X y}.
Let X0,X1,... be a Markov chain whose state space is Z (the
integers).
Recall the Markov property: P(Xn = in | X0 = i0,X1 = i1,...,Xn−1
= in−1) = P(Xn = in | Xn−1 = in−1), ∀n, ∀it. Does the following
always hold: P(Xn ≥0|X0 ≥0,X1 ≥0,...,Xn−1 ≥0)=P(Xn ≥0|Xn−1 ≥0)
?
(Prove if “yes”, provide a counterexample if “no”)
Let Xo,Xi, be a Markov chain whose state space is Z (the integers). Recall the Markov property: P(X,-'n l Xo-io, Xi...
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that T has a geometric distribution with respect to the conditional probability P
1. Exit times. Let X be a discrete-time Markov chain (with discrete state space) and suppose pii > 0. Let T =min{n 21: X i} be the exit time from state i. Show that...
1. Let {Xn, n 2 0 be a Markov Chain with state space S. Show that for any n, m-1 and JAn+m, . . . , İn+1,in-1 , . . . , io є S.
4. Fix > 0. For n > λ let Xn be Geometric(A/n). Show that X n/n converges in distribution to an Exponential(A). (Hint: again, compute moment generating functions.)
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) = E (eitX»), ER be their characteristic functions. Let Yn = {k=0 Xk, n > 0, X0 = 0, and 8. () = {1*: (),ER. k = 1 a) Let t be so that I1=1 løk (t)) > 0. Show that _exp{itYn} ?, n > 0, On (t) is a martingale with respect to Fn = (Xo, ...,Xn), n > 0, and sup, E (M,|2)...
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