Let Ek, k > 0, be independent random variables taking only the integers. Let S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable, independent of the sequence &k, taking values in S and let ƒ : S × Z → S be a certain real function. Define new random variables X,„ by Xn+1 = f(Xn, En), n = 0, 1,2.. (a) Show that {X„}n21 is a Markov chain with state space S = {1, ..., N} and this chain is homogeneous if only §k, k > 1, have the same distribution.
Let Ek, k > 0, be independent random variables taking only the integers. Let S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable, independent of the sequence &k, taking values in S and let ƒ : S × Z → S be a certain real function. Define new random variables X,„ by Xn+1 = f(Xn, En), n = 0, 1,2.. (a) Show that {X„}n21 is a Markov chain with state space S = {1, ..., N} and this chain is homogeneous if only §k, k > 1, have the same distribution.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![Let Ek, k > 0, be independent random variables taking only the integers. Let
S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable,
independent of the sequence &k, taking values in S and let f : S x Z → S be a
certain real function. Define new random variables X, by
Xn+1
f(Xn, §n), n = 0, 1, 2...
(a) Show that {X„}n21 is a Markov chain with state space S = {1, .., N} and this
chain is homogeneous if only Fk, k > 1, have the same distribution.
(b) Suppose that Xo = 0, f(x, y) = x+y and fk, k > 0, are iid with P({o = 1) = p,
P(5o = 0) =r and P(£o = -1) = q. where p,r, q > 0 and p+r+q = 1.
i. Write down the transition probabilities of the chain {Xn}n>1-
ii. Is the chain {Xn}n>1 aperiodic and irreducible?
iii. Find expressions for:
P(X3 = 2), P(X4 = 1|X1 = 1) and P(X10 = 1|X7 = 0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4228311-ae76-4ab0-9152-6ef10b2cc34d%2Fa64de897-eb0b-4f63-ae98-75e475e2adc7%2Fyd53hq_processed.png&w=3840&q=75)
Transcribed Image Text:Let Ek, k > 0, be independent random variables taking only the integers. Let
S = {1,2, ..., N}, where N = ∞ is allowed. Let Xo be another random variable,
independent of the sequence &k, taking values in S and let f : S x Z → S be a
certain real function. Define new random variables X, by
Xn+1
f(Xn, §n), n = 0, 1, 2...
(a) Show that {X„}n21 is a Markov chain with state space S = {1, .., N} and this
chain is homogeneous if only Fk, k > 1, have the same distribution.
(b) Suppose that Xo = 0, f(x, y) = x+y and fk, k > 0, are iid with P({o = 1) = p,
P(5o = 0) =r and P(£o = -1) = q. where p,r, q > 0 and p+r+q = 1.
i. Write down the transition probabilities of the chain {Xn}n>1-
ii. Is the chain {Xn}n>1 aperiodic and irreducible?
iii. Find expressions for:
P(X3 = 2), P(X4 = 1|X1 = 1) and P(X10 = 1|X7 = 0).
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