Proof of homogeneous Markov chain and state-transition matrix
Let $X_i$ be the results of a consecutive throws of a dice. Let $Z_n=3(X_1^2+...+X_n^2) mod 5$. Show that the sequence ${{Z_n | ngeq1}}$ is a homogeneous Markov Chain. Find a state space and a state-trasition matrix.
I know the Markov chain property which is
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1}) = P(X_n=s|X_{n-1}=x_{n-1})$ and then the Markov chain is homogeneous, when
$P(X_{n+1}=j|X_n=i)=P(X_1=j|X_0=i)$
We know that $X_1,...,X_n$ are independent so
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1})=P(X_n=s)$ and $P(X_n=s|X_{n-1}=x_{n-1})=P(X_n=s)$ hence $P(X_{n+1}=j)=P(X_1=j)$,
but from this point I don't know how to continue.
Any help will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
add a comment |
Let $X_i$ be the results of a consecutive throws of a dice. Let $Z_n=3(X_1^2+...+X_n^2) mod 5$. Show that the sequence ${{Z_n | ngeq1}}$ is a homogeneous Markov Chain. Find a state space and a state-trasition matrix.
I know the Markov chain property which is
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1}) = P(X_n=s|X_{n-1}=x_{n-1})$ and then the Markov chain is homogeneous, when
$P(X_{n+1}=j|X_n=i)=P(X_1=j|X_0=i)$
We know that $X_1,...,X_n$ are independent so
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1})=P(X_n=s)$ and $P(X_n=s|X_{n-1}=x_{n-1})=P(X_n=s)$ hence $P(X_{n+1}=j)=P(X_1=j)$,
but from this point I don't know how to continue.
Any help will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
add a comment |
Let $X_i$ be the results of a consecutive throws of a dice. Let $Z_n=3(X_1^2+...+X_n^2) mod 5$. Show that the sequence ${{Z_n | ngeq1}}$ is a homogeneous Markov Chain. Find a state space and a state-trasition matrix.
I know the Markov chain property which is
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1}) = P(X_n=s|X_{n-1}=x_{n-1})$ and then the Markov chain is homogeneous, when
$P(X_{n+1}=j|X_n=i)=P(X_1=j|X_0=i)$
We know that $X_1,...,X_n$ are independent so
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1})=P(X_n=s)$ and $P(X_n=s|X_{n-1}=x_{n-1})=P(X_n=s)$ hence $P(X_{n+1}=j)=P(X_1=j)$,
but from this point I don't know how to continue.
Any help will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
Let $X_i$ be the results of a consecutive throws of a dice. Let $Z_n=3(X_1^2+...+X_n^2) mod 5$. Show that the sequence ${{Z_n | ngeq1}}$ is a homogeneous Markov Chain. Find a state space and a state-trasition matrix.
I know the Markov chain property which is
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1}) = P(X_n=s|X_{n-1}=x_{n-1})$ and then the Markov chain is homogeneous, when
$P(X_{n+1}=j|X_n=i)=P(X_1=j|X_0=i)$
We know that $X_1,...,X_n$ are independent so
$P(X_n=s | X_0=x_0,...,X_{n-1}=x_{n-1})=P(X_n=s)$ and $P(X_n=s|X_{n-1}=x_{n-1})=P(X_n=s)$ hence $P(X_{n+1}=j)=P(X_1=j)$,
but from this point I don't know how to continue.
Any help will be much appreciated.
probability-theory stochastic-processes markov-chains
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
edited 2 days ago
MacAbra
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
asked Dec 15 '18 at 12:25
MacAbraMacAbra
17519
17519
add a comment |
add a comment |
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