Expected value and variance of step of random walk with barriers











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I have to simulate the following game




Suppose that two players A and B each start with a stake of $5, and bet $0.5 on consecutive coin flips. The game ends when either one of the players has won all the money, that amounts to $10. Let $S_n$ be the fortune of player A at time n. Then ${S_n, n gt 0 }$ is a symmetric random walk with absorbing barriers at 0 and 10. Estimate $E[S_n]$ and $V[S_n]$ for $n = 50$.




I made a program and that's ok. Now I would to compare my results with the theoretical values. I don't know almost anything in probability, so that's just a curiosity. My question is




Which are the values of $E[S_n]$ and $V[S_n]$ in general? And for $n = 50$?




If there's no closed form, I will appreciate also an approximation (I got $E[S_n] approx 5$ and $V[S_{50}] approx 4.8$)



Thanks in advance.










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  • If you have simulated, I'm sure you can guess the (very simple) value of $E[S_n]$. The proper way to simulate that is to run multiple (say, 10000 or more) experiments, get a time series for each experiment $i$ to obtain the vector $(S_0^{(i)}, S_1^{(i)}, S_2^{(i)}, ..., S_{100}^{(i)})$, average those values and plot the averaged values $frac{1}{10000}sum_{i=1}^{10000}S_n^{(i)}$ versus $n in {0, 1, 2, ..., 100}$.
    – Michael
    Nov 22 at 15:25












  • Yes I did, but now I want to know the theoretical results
    – Marco All-in Nervo
    Nov 22 at 15:34










  • So from those experiments you should be able to guess the value of $E[S_n]$ for each $n$ and your guess is...
    – Michael
    Nov 22 at 15:34










  • My guess is 5 but I want to know if it agrees with the theory
    – Marco All-in Nervo
    Nov 22 at 15:56










  • Yes $E[S_n]=5$ for all $n$ since this is a martingale, $S_{n+1} = S_n + A_n$ where $A_n=0$ if $S_n in {0, 10}$ and $E[A_n|S_n]=0$ regardless the value of $S_n$. You might want to provide your simulated guesses for $E[S_{50}]$ and $Var(S_{50})$ in the question itself. Here I assume coin flips are independent and equally likely to be heads or tails.
    – Michael
    Nov 22 at 16:02

















up vote
2
down vote

favorite












I have to simulate the following game




Suppose that two players A and B each start with a stake of $5, and bet $0.5 on consecutive coin flips. The game ends when either one of the players has won all the money, that amounts to $10. Let $S_n$ be the fortune of player A at time n. Then ${S_n, n gt 0 }$ is a symmetric random walk with absorbing barriers at 0 and 10. Estimate $E[S_n]$ and $V[S_n]$ for $n = 50$.




I made a program and that's ok. Now I would to compare my results with the theoretical values. I don't know almost anything in probability, so that's just a curiosity. My question is




Which are the values of $E[S_n]$ and $V[S_n]$ in general? And for $n = 50$?




If there's no closed form, I will appreciate also an approximation (I got $E[S_n] approx 5$ and $V[S_{50}] approx 4.8$)



Thanks in advance.










share|cite|improve this question
























  • If you have simulated, I'm sure you can guess the (very simple) value of $E[S_n]$. The proper way to simulate that is to run multiple (say, 10000 or more) experiments, get a time series for each experiment $i$ to obtain the vector $(S_0^{(i)}, S_1^{(i)}, S_2^{(i)}, ..., S_{100}^{(i)})$, average those values and plot the averaged values $frac{1}{10000}sum_{i=1}^{10000}S_n^{(i)}$ versus $n in {0, 1, 2, ..., 100}$.
    – Michael
    Nov 22 at 15:25












  • Yes I did, but now I want to know the theoretical results
    – Marco All-in Nervo
    Nov 22 at 15:34










  • So from those experiments you should be able to guess the value of $E[S_n]$ for each $n$ and your guess is...
    – Michael
    Nov 22 at 15:34










  • My guess is 5 but I want to know if it agrees with the theory
    – Marco All-in Nervo
    Nov 22 at 15:56










  • Yes $E[S_n]=5$ for all $n$ since this is a martingale, $S_{n+1} = S_n + A_n$ where $A_n=0$ if $S_n in {0, 10}$ and $E[A_n|S_n]=0$ regardless the value of $S_n$. You might want to provide your simulated guesses for $E[S_{50}]$ and $Var(S_{50})$ in the question itself. Here I assume coin flips are independent and equally likely to be heads or tails.
    – Michael
    Nov 22 at 16:02















up vote
2
down vote

favorite









up vote
2
down vote

favorite











I have to simulate the following game




Suppose that two players A and B each start with a stake of $5, and bet $0.5 on consecutive coin flips. The game ends when either one of the players has won all the money, that amounts to $10. Let $S_n$ be the fortune of player A at time n. Then ${S_n, n gt 0 }$ is a symmetric random walk with absorbing barriers at 0 and 10. Estimate $E[S_n]$ and $V[S_n]$ for $n = 50$.




I made a program and that's ok. Now I would to compare my results with the theoretical values. I don't know almost anything in probability, so that's just a curiosity. My question is




Which are the values of $E[S_n]$ and $V[S_n]$ in general? And for $n = 50$?




If there's no closed form, I will appreciate also an approximation (I got $E[S_n] approx 5$ and $V[S_{50}] approx 4.8$)



Thanks in advance.










share|cite|improve this question















I have to simulate the following game




Suppose that two players A and B each start with a stake of $5, and bet $0.5 on consecutive coin flips. The game ends when either one of the players has won all the money, that amounts to $10. Let $S_n$ be the fortune of player A at time n. Then ${S_n, n gt 0 }$ is a symmetric random walk with absorbing barriers at 0 and 10. Estimate $E[S_n]$ and $V[S_n]$ for $n = 50$.




I made a program and that's ok. Now I would to compare my results with the theoretical values. I don't know almost anything in probability, so that's just a curiosity. My question is




Which are the values of $E[S_n]$ and $V[S_n]$ in general? And for $n = 50$?




If there's no closed form, I will appreciate also an approximation (I got $E[S_n] approx 5$ and $V[S_{50}] approx 4.8$)



Thanks in advance.







probability random-walk means






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edited Nov 22 at 16:25

























asked Nov 22 at 14:57









Marco All-in Nervo

34918




34918












  • If you have simulated, I'm sure you can guess the (very simple) value of $E[S_n]$. The proper way to simulate that is to run multiple (say, 10000 or more) experiments, get a time series for each experiment $i$ to obtain the vector $(S_0^{(i)}, S_1^{(i)}, S_2^{(i)}, ..., S_{100}^{(i)})$, average those values and plot the averaged values $frac{1}{10000}sum_{i=1}^{10000}S_n^{(i)}$ versus $n in {0, 1, 2, ..., 100}$.
    – Michael
    Nov 22 at 15:25












  • Yes I did, but now I want to know the theoretical results
    – Marco All-in Nervo
    Nov 22 at 15:34










  • So from those experiments you should be able to guess the value of $E[S_n]$ for each $n$ and your guess is...
    – Michael
    Nov 22 at 15:34










  • My guess is 5 but I want to know if it agrees with the theory
    – Marco All-in Nervo
    Nov 22 at 15:56










  • Yes $E[S_n]=5$ for all $n$ since this is a martingale, $S_{n+1} = S_n + A_n$ where $A_n=0$ if $S_n in {0, 10}$ and $E[A_n|S_n]=0$ regardless the value of $S_n$. You might want to provide your simulated guesses for $E[S_{50}]$ and $Var(S_{50})$ in the question itself. Here I assume coin flips are independent and equally likely to be heads or tails.
    – Michael
    Nov 22 at 16:02




















  • If you have simulated, I'm sure you can guess the (very simple) value of $E[S_n]$. The proper way to simulate that is to run multiple (say, 10000 or more) experiments, get a time series for each experiment $i$ to obtain the vector $(S_0^{(i)}, S_1^{(i)}, S_2^{(i)}, ..., S_{100}^{(i)})$, average those values and plot the averaged values $frac{1}{10000}sum_{i=1}^{10000}S_n^{(i)}$ versus $n in {0, 1, 2, ..., 100}$.
    – Michael
    Nov 22 at 15:25












  • Yes I did, but now I want to know the theoretical results
    – Marco All-in Nervo
    Nov 22 at 15:34










  • So from those experiments you should be able to guess the value of $E[S_n]$ for each $n$ and your guess is...
    – Michael
    Nov 22 at 15:34










  • My guess is 5 but I want to know if it agrees with the theory
    – Marco All-in Nervo
    Nov 22 at 15:56










  • Yes $E[S_n]=5$ for all $n$ since this is a martingale, $S_{n+1} = S_n + A_n$ where $A_n=0$ if $S_n in {0, 10}$ and $E[A_n|S_n]=0$ regardless the value of $S_n$. You might want to provide your simulated guesses for $E[S_{50}]$ and $Var(S_{50})$ in the question itself. Here I assume coin flips are independent and equally likely to be heads or tails.
    – Michael
    Nov 22 at 16:02


















If you have simulated, I'm sure you can guess the (very simple) value of $E[S_n]$. The proper way to simulate that is to run multiple (say, 10000 or more) experiments, get a time series for each experiment $i$ to obtain the vector $(S_0^{(i)}, S_1^{(i)}, S_2^{(i)}, ..., S_{100}^{(i)})$, average those values and plot the averaged values $frac{1}{10000}sum_{i=1}^{10000}S_n^{(i)}$ versus $n in {0, 1, 2, ..., 100}$.
– Michael
Nov 22 at 15:25






If you have simulated, I'm sure you can guess the (very simple) value of $E[S_n]$. The proper way to simulate that is to run multiple (say, 10000 or more) experiments, get a time series for each experiment $i$ to obtain the vector $(S_0^{(i)}, S_1^{(i)}, S_2^{(i)}, ..., S_{100}^{(i)})$, average those values and plot the averaged values $frac{1}{10000}sum_{i=1}^{10000}S_n^{(i)}$ versus $n in {0, 1, 2, ..., 100}$.
– Michael
Nov 22 at 15:25














Yes I did, but now I want to know the theoretical results
– Marco All-in Nervo
Nov 22 at 15:34




Yes I did, but now I want to know the theoretical results
– Marco All-in Nervo
Nov 22 at 15:34












So from those experiments you should be able to guess the value of $E[S_n]$ for each $n$ and your guess is...
– Michael
Nov 22 at 15:34




So from those experiments you should be able to guess the value of $E[S_n]$ for each $n$ and your guess is...
– Michael
Nov 22 at 15:34












My guess is 5 but I want to know if it agrees with the theory
– Marco All-in Nervo
Nov 22 at 15:56




My guess is 5 but I want to know if it agrees with the theory
– Marco All-in Nervo
Nov 22 at 15:56












Yes $E[S_n]=5$ for all $n$ since this is a martingale, $S_{n+1} = S_n + A_n$ where $A_n=0$ if $S_n in {0, 10}$ and $E[A_n|S_n]=0$ regardless the value of $S_n$. You might want to provide your simulated guesses for $E[S_{50}]$ and $Var(S_{50})$ in the question itself. Here I assume coin flips are independent and equally likely to be heads or tails.
– Michael
Nov 22 at 16:02






Yes $E[S_n]=5$ for all $n$ since this is a martingale, $S_{n+1} = S_n + A_n$ where $A_n=0$ if $S_n in {0, 10}$ and $E[A_n|S_n]=0$ regardless the value of $S_n$. You might want to provide your simulated guesses for $E[S_{50}]$ and $Var(S_{50})$ in the question itself. Here I assume coin flips are independent and equally likely to be heads or tails.
– Michael
Nov 22 at 16:02












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Modeling the process



Define $mathcal{S}$ as the set of possible values for the Markov chain:
$$mathcal{S} = {0, 0.5, 1, 1.5, …, 9.5, 10}$$
Note that $S_0=5$ and $S_n in mathcal{S}$ for all $n in {0, 1, 2, …}$.
We have
$$S_{n+1} = S_n + A_n quad forall n in {0, 1, 2, ...} $$
where
$$ A_n = left{ begin{array}{ll}
(1/2)B_n &mbox{ if $S_n notin {0, 10}$} \
0 & mbox{ otherwise}
end{array}
right.$$

where ${B_n}_{n=0}^{infty}$ is an i.i.d. sequence with $P[B_n=1]=P[B_n=-1]=1/2$.
Then
$$boxed{E[A_n|S_n=s] = 0 quad, forall s in mathcal{S}} quad (Eq. 1) $$





Mean



So for each $n in {0, 1, 2, ...}$ we have
begin{align}
E[S_{n+1}] &overset{(a)}{=} sum_{s in mathcal{S}}E[S_{n+1}|S_n=s]P[S_n=s] \
&overset{(b)}{=} sum_{s in mathcal{S}}E[S_n + A_n|S_n=s]P[S_n=s] \
&= sum_{s in mathcal{S}}E[s + A_n|S_n=s]P[S_n=s] \
&= sum_{s in mathcal{S}}(s + E[A_n|S_n=s])P[S_n=s] \
&overset{(c)}{=} sum_{s in mathcal{S}}sP[S_n=s] \
&overset{(d)}{=} E[S_n]
end{align}

where (a) holds by the law of total expectation; (b) holds by the fact $S_{n+1}=S_n+A_n$; (c) holds by Eq. (1); (d) holds by definition of expectation.
Since $E[S_0]=5$ we conclude:
$$boxed{E[S_n]=5 quad forall n in {0, 1, 2, … }}$$





Limiting variance



We know $E[S_n]=5$ for all $n$ and so
$$Var(S_n) = E[(S_n-5)^2] = sum_{s in mathcal{S}}(s-5)^2P[S_n=s] $$
Since the process is equally likely to end up at state $0$ or $10$ we have
begin{align}
lim_{nrightarrowinfty} P[S_n=0] &= 1/2\
lim_{nrightarrowinfty} P[S_n=10] &= 1/2\
lim_{nrightarrowinfty} P[S_n=s] &= 0 quad forall s notin {0, 10}
end{align}

so
$$ boxed{lim_{nrightarrowinfty} Var(S_n) = (0-5)^2(1/2) + (10-5)^2(1/2) = 25} $$





Details on variance



Squaring the equation $S_{n+1} = S_n + A_n$ gives
$$S_{n+1}^2 = (S_n+A_n)^2 = S_n^2 + 2S_nA_n + A_n^2 $$
So
$$E[S_{n+1}^2|S_n] = S_n^2 + 2S_nE[A_n|S_n] + E[A_n^2|S_n] = S_n^2 + 0 + (1/4)1_{{S_n notin{0, 10}}}$$
where $1_{{S_n notin{0, 10}}}$ is an indicator function that is 1 if $S_n notin {0,10}$ and is 0 else. So
$$E[S_{n+1}^2] = E[S_n^2] + (1/4)P[S_n notin {0,10}]$$
Subtracting 25 from both sides gives
$$ Var(S_{n+1}) = Var(S_n) + (1/4)P[S_n notin {0,10}]$$
and $Var(S_0)=0$ so
$$ boxed{Var(S_n) = (1/4)sum_{i=0}^{n-1} P[S_i notin {0,10}] quad forall nin {1, 2, 3, ...} } $$
Since $P[S_i notin {0,10}] = 1$ for $i in {0, 1, 2, 3, ..., 9}$ we have $$boxed{Var(S_1)=1/4, Var(S_2)=2/4, Var(S_3) = 3/4, ..., Var(S_{10})= 10/4}$$
On the other hand:
$$ Var(S_{11}) = 10/4 + (1/4)underbrace{(1-2(1/2)^{10})}_{P[S_{10}notin{0,10}]}$$
In general, the variance increases as $nrightarrowinfty$ to approach a
limiting value of $25$. It is possible to compute $P[S_i notin {0,10}]$ for all $i$ (for example, by taking powers of a transition probability matrix), but this calculation is more involved.






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    down vote



    accepted










    Modeling the process



    Define $mathcal{S}$ as the set of possible values for the Markov chain:
    $$mathcal{S} = {0, 0.5, 1, 1.5, …, 9.5, 10}$$
    Note that $S_0=5$ and $S_n in mathcal{S}$ for all $n in {0, 1, 2, …}$.
    We have
    $$S_{n+1} = S_n + A_n quad forall n in {0, 1, 2, ...} $$
    where
    $$ A_n = left{ begin{array}{ll}
    (1/2)B_n &mbox{ if $S_n notin {0, 10}$} \
    0 & mbox{ otherwise}
    end{array}
    right.$$

    where ${B_n}_{n=0}^{infty}$ is an i.i.d. sequence with $P[B_n=1]=P[B_n=-1]=1/2$.
    Then
    $$boxed{E[A_n|S_n=s] = 0 quad, forall s in mathcal{S}} quad (Eq. 1) $$





    Mean



    So for each $n in {0, 1, 2, ...}$ we have
    begin{align}
    E[S_{n+1}] &overset{(a)}{=} sum_{s in mathcal{S}}E[S_{n+1}|S_n=s]P[S_n=s] \
    &overset{(b)}{=} sum_{s in mathcal{S}}E[S_n + A_n|S_n=s]P[S_n=s] \
    &= sum_{s in mathcal{S}}E[s + A_n|S_n=s]P[S_n=s] \
    &= sum_{s in mathcal{S}}(s + E[A_n|S_n=s])P[S_n=s] \
    &overset{(c)}{=} sum_{s in mathcal{S}}sP[S_n=s] \
    &overset{(d)}{=} E[S_n]
    end{align}

    where (a) holds by the law of total expectation; (b) holds by the fact $S_{n+1}=S_n+A_n$; (c) holds by Eq. (1); (d) holds by definition of expectation.
    Since $E[S_0]=5$ we conclude:
    $$boxed{E[S_n]=5 quad forall n in {0, 1, 2, … }}$$





    Limiting variance



    We know $E[S_n]=5$ for all $n$ and so
    $$Var(S_n) = E[(S_n-5)^2] = sum_{s in mathcal{S}}(s-5)^2P[S_n=s] $$
    Since the process is equally likely to end up at state $0$ or $10$ we have
    begin{align}
    lim_{nrightarrowinfty} P[S_n=0] &= 1/2\
    lim_{nrightarrowinfty} P[S_n=10] &= 1/2\
    lim_{nrightarrowinfty} P[S_n=s] &= 0 quad forall s notin {0, 10}
    end{align}

    so
    $$ boxed{lim_{nrightarrowinfty} Var(S_n) = (0-5)^2(1/2) + (10-5)^2(1/2) = 25} $$





    Details on variance



    Squaring the equation $S_{n+1} = S_n + A_n$ gives
    $$S_{n+1}^2 = (S_n+A_n)^2 = S_n^2 + 2S_nA_n + A_n^2 $$
    So
    $$E[S_{n+1}^2|S_n] = S_n^2 + 2S_nE[A_n|S_n] + E[A_n^2|S_n] = S_n^2 + 0 + (1/4)1_{{S_n notin{0, 10}}}$$
    where $1_{{S_n notin{0, 10}}}$ is an indicator function that is 1 if $S_n notin {0,10}$ and is 0 else. So
    $$E[S_{n+1}^2] = E[S_n^2] + (1/4)P[S_n notin {0,10}]$$
    Subtracting 25 from both sides gives
    $$ Var(S_{n+1}) = Var(S_n) + (1/4)P[S_n notin {0,10}]$$
    and $Var(S_0)=0$ so
    $$ boxed{Var(S_n) = (1/4)sum_{i=0}^{n-1} P[S_i notin {0,10}] quad forall nin {1, 2, 3, ...} } $$
    Since $P[S_i notin {0,10}] = 1$ for $i in {0, 1, 2, 3, ..., 9}$ we have $$boxed{Var(S_1)=1/4, Var(S_2)=2/4, Var(S_3) = 3/4, ..., Var(S_{10})= 10/4}$$
    On the other hand:
    $$ Var(S_{11}) = 10/4 + (1/4)underbrace{(1-2(1/2)^{10})}_{P[S_{10}notin{0,10}]}$$
    In general, the variance increases as $nrightarrowinfty$ to approach a
    limiting value of $25$. It is possible to compute $P[S_i notin {0,10}]$ for all $i$ (for example, by taking powers of a transition probability matrix), but this calculation is more involved.






    share|cite|improve this answer



























      up vote
      1
      down vote



      accepted










      Modeling the process



      Define $mathcal{S}$ as the set of possible values for the Markov chain:
      $$mathcal{S} = {0, 0.5, 1, 1.5, …, 9.5, 10}$$
      Note that $S_0=5$ and $S_n in mathcal{S}$ for all $n in {0, 1, 2, …}$.
      We have
      $$S_{n+1} = S_n + A_n quad forall n in {0, 1, 2, ...} $$
      where
      $$ A_n = left{ begin{array}{ll}
      (1/2)B_n &mbox{ if $S_n notin {0, 10}$} \
      0 & mbox{ otherwise}
      end{array}
      right.$$

      where ${B_n}_{n=0}^{infty}$ is an i.i.d. sequence with $P[B_n=1]=P[B_n=-1]=1/2$.
      Then
      $$boxed{E[A_n|S_n=s] = 0 quad, forall s in mathcal{S}} quad (Eq. 1) $$





      Mean



      So for each $n in {0, 1, 2, ...}$ we have
      begin{align}
      E[S_{n+1}] &overset{(a)}{=} sum_{s in mathcal{S}}E[S_{n+1}|S_n=s]P[S_n=s] \
      &overset{(b)}{=} sum_{s in mathcal{S}}E[S_n + A_n|S_n=s]P[S_n=s] \
      &= sum_{s in mathcal{S}}E[s + A_n|S_n=s]P[S_n=s] \
      &= sum_{s in mathcal{S}}(s + E[A_n|S_n=s])P[S_n=s] \
      &overset{(c)}{=} sum_{s in mathcal{S}}sP[S_n=s] \
      &overset{(d)}{=} E[S_n]
      end{align}

      where (a) holds by the law of total expectation; (b) holds by the fact $S_{n+1}=S_n+A_n$; (c) holds by Eq. (1); (d) holds by definition of expectation.
      Since $E[S_0]=5$ we conclude:
      $$boxed{E[S_n]=5 quad forall n in {0, 1, 2, … }}$$





      Limiting variance



      We know $E[S_n]=5$ for all $n$ and so
      $$Var(S_n) = E[(S_n-5)^2] = sum_{s in mathcal{S}}(s-5)^2P[S_n=s] $$
      Since the process is equally likely to end up at state $0$ or $10$ we have
      begin{align}
      lim_{nrightarrowinfty} P[S_n=0] &= 1/2\
      lim_{nrightarrowinfty} P[S_n=10] &= 1/2\
      lim_{nrightarrowinfty} P[S_n=s] &= 0 quad forall s notin {0, 10}
      end{align}

      so
      $$ boxed{lim_{nrightarrowinfty} Var(S_n) = (0-5)^2(1/2) + (10-5)^2(1/2) = 25} $$





      Details on variance



      Squaring the equation $S_{n+1} = S_n + A_n$ gives
      $$S_{n+1}^2 = (S_n+A_n)^2 = S_n^2 + 2S_nA_n + A_n^2 $$
      So
      $$E[S_{n+1}^2|S_n] = S_n^2 + 2S_nE[A_n|S_n] + E[A_n^2|S_n] = S_n^2 + 0 + (1/4)1_{{S_n notin{0, 10}}}$$
      where $1_{{S_n notin{0, 10}}}$ is an indicator function that is 1 if $S_n notin {0,10}$ and is 0 else. So
      $$E[S_{n+1}^2] = E[S_n^2] + (1/4)P[S_n notin {0,10}]$$
      Subtracting 25 from both sides gives
      $$ Var(S_{n+1}) = Var(S_n) + (1/4)P[S_n notin {0,10}]$$
      and $Var(S_0)=0$ so
      $$ boxed{Var(S_n) = (1/4)sum_{i=0}^{n-1} P[S_i notin {0,10}] quad forall nin {1, 2, 3, ...} } $$
      Since $P[S_i notin {0,10}] = 1$ for $i in {0, 1, 2, 3, ..., 9}$ we have $$boxed{Var(S_1)=1/4, Var(S_2)=2/4, Var(S_3) = 3/4, ..., Var(S_{10})= 10/4}$$
      On the other hand:
      $$ Var(S_{11}) = 10/4 + (1/4)underbrace{(1-2(1/2)^{10})}_{P[S_{10}notin{0,10}]}$$
      In general, the variance increases as $nrightarrowinfty$ to approach a
      limiting value of $25$. It is possible to compute $P[S_i notin {0,10}]$ for all $i$ (for example, by taking powers of a transition probability matrix), but this calculation is more involved.






      share|cite|improve this answer

























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        accepted






        Modeling the process



        Define $mathcal{S}$ as the set of possible values for the Markov chain:
        $$mathcal{S} = {0, 0.5, 1, 1.5, …, 9.5, 10}$$
        Note that $S_0=5$ and $S_n in mathcal{S}$ for all $n in {0, 1, 2, …}$.
        We have
        $$S_{n+1} = S_n + A_n quad forall n in {0, 1, 2, ...} $$
        where
        $$ A_n = left{ begin{array}{ll}
        (1/2)B_n &mbox{ if $S_n notin {0, 10}$} \
        0 & mbox{ otherwise}
        end{array}
        right.$$

        where ${B_n}_{n=0}^{infty}$ is an i.i.d. sequence with $P[B_n=1]=P[B_n=-1]=1/2$.
        Then
        $$boxed{E[A_n|S_n=s] = 0 quad, forall s in mathcal{S}} quad (Eq. 1) $$





        Mean



        So for each $n in {0, 1, 2, ...}$ we have
        begin{align}
        E[S_{n+1}] &overset{(a)}{=} sum_{s in mathcal{S}}E[S_{n+1}|S_n=s]P[S_n=s] \
        &overset{(b)}{=} sum_{s in mathcal{S}}E[S_n + A_n|S_n=s]P[S_n=s] \
        &= sum_{s in mathcal{S}}E[s + A_n|S_n=s]P[S_n=s] \
        &= sum_{s in mathcal{S}}(s + E[A_n|S_n=s])P[S_n=s] \
        &overset{(c)}{=} sum_{s in mathcal{S}}sP[S_n=s] \
        &overset{(d)}{=} E[S_n]
        end{align}

        where (a) holds by the law of total expectation; (b) holds by the fact $S_{n+1}=S_n+A_n$; (c) holds by Eq. (1); (d) holds by definition of expectation.
        Since $E[S_0]=5$ we conclude:
        $$boxed{E[S_n]=5 quad forall n in {0, 1, 2, … }}$$





        Limiting variance



        We know $E[S_n]=5$ for all $n$ and so
        $$Var(S_n) = E[(S_n-5)^2] = sum_{s in mathcal{S}}(s-5)^2P[S_n=s] $$
        Since the process is equally likely to end up at state $0$ or $10$ we have
        begin{align}
        lim_{nrightarrowinfty} P[S_n=0] &= 1/2\
        lim_{nrightarrowinfty} P[S_n=10] &= 1/2\
        lim_{nrightarrowinfty} P[S_n=s] &= 0 quad forall s notin {0, 10}
        end{align}

        so
        $$ boxed{lim_{nrightarrowinfty} Var(S_n) = (0-5)^2(1/2) + (10-5)^2(1/2) = 25} $$





        Details on variance



        Squaring the equation $S_{n+1} = S_n + A_n$ gives
        $$S_{n+1}^2 = (S_n+A_n)^2 = S_n^2 + 2S_nA_n + A_n^2 $$
        So
        $$E[S_{n+1}^2|S_n] = S_n^2 + 2S_nE[A_n|S_n] + E[A_n^2|S_n] = S_n^2 + 0 + (1/4)1_{{S_n notin{0, 10}}}$$
        where $1_{{S_n notin{0, 10}}}$ is an indicator function that is 1 if $S_n notin {0,10}$ and is 0 else. So
        $$E[S_{n+1}^2] = E[S_n^2] + (1/4)P[S_n notin {0,10}]$$
        Subtracting 25 from both sides gives
        $$ Var(S_{n+1}) = Var(S_n) + (1/4)P[S_n notin {0,10}]$$
        and $Var(S_0)=0$ so
        $$ boxed{Var(S_n) = (1/4)sum_{i=0}^{n-1} P[S_i notin {0,10}] quad forall nin {1, 2, 3, ...} } $$
        Since $P[S_i notin {0,10}] = 1$ for $i in {0, 1, 2, 3, ..., 9}$ we have $$boxed{Var(S_1)=1/4, Var(S_2)=2/4, Var(S_3) = 3/4, ..., Var(S_{10})= 10/4}$$
        On the other hand:
        $$ Var(S_{11}) = 10/4 + (1/4)underbrace{(1-2(1/2)^{10})}_{P[S_{10}notin{0,10}]}$$
        In general, the variance increases as $nrightarrowinfty$ to approach a
        limiting value of $25$. It is possible to compute $P[S_i notin {0,10}]$ for all $i$ (for example, by taking powers of a transition probability matrix), but this calculation is more involved.






        share|cite|improve this answer














        Modeling the process



        Define $mathcal{S}$ as the set of possible values for the Markov chain:
        $$mathcal{S} = {0, 0.5, 1, 1.5, …, 9.5, 10}$$
        Note that $S_0=5$ and $S_n in mathcal{S}$ for all $n in {0, 1, 2, …}$.
        We have
        $$S_{n+1} = S_n + A_n quad forall n in {0, 1, 2, ...} $$
        where
        $$ A_n = left{ begin{array}{ll}
        (1/2)B_n &mbox{ if $S_n notin {0, 10}$} \
        0 & mbox{ otherwise}
        end{array}
        right.$$

        where ${B_n}_{n=0}^{infty}$ is an i.i.d. sequence with $P[B_n=1]=P[B_n=-1]=1/2$.
        Then
        $$boxed{E[A_n|S_n=s] = 0 quad, forall s in mathcal{S}} quad (Eq. 1) $$





        Mean



        So for each $n in {0, 1, 2, ...}$ we have
        begin{align}
        E[S_{n+1}] &overset{(a)}{=} sum_{s in mathcal{S}}E[S_{n+1}|S_n=s]P[S_n=s] \
        &overset{(b)}{=} sum_{s in mathcal{S}}E[S_n + A_n|S_n=s]P[S_n=s] \
        &= sum_{s in mathcal{S}}E[s + A_n|S_n=s]P[S_n=s] \
        &= sum_{s in mathcal{S}}(s + E[A_n|S_n=s])P[S_n=s] \
        &overset{(c)}{=} sum_{s in mathcal{S}}sP[S_n=s] \
        &overset{(d)}{=} E[S_n]
        end{align}

        where (a) holds by the law of total expectation; (b) holds by the fact $S_{n+1}=S_n+A_n$; (c) holds by Eq. (1); (d) holds by definition of expectation.
        Since $E[S_0]=5$ we conclude:
        $$boxed{E[S_n]=5 quad forall n in {0, 1, 2, … }}$$





        Limiting variance



        We know $E[S_n]=5$ for all $n$ and so
        $$Var(S_n) = E[(S_n-5)^2] = sum_{s in mathcal{S}}(s-5)^2P[S_n=s] $$
        Since the process is equally likely to end up at state $0$ or $10$ we have
        begin{align}
        lim_{nrightarrowinfty} P[S_n=0] &= 1/2\
        lim_{nrightarrowinfty} P[S_n=10] &= 1/2\
        lim_{nrightarrowinfty} P[S_n=s] &= 0 quad forall s notin {0, 10}
        end{align}

        so
        $$ boxed{lim_{nrightarrowinfty} Var(S_n) = (0-5)^2(1/2) + (10-5)^2(1/2) = 25} $$





        Details on variance



        Squaring the equation $S_{n+1} = S_n + A_n$ gives
        $$S_{n+1}^2 = (S_n+A_n)^2 = S_n^2 + 2S_nA_n + A_n^2 $$
        So
        $$E[S_{n+1}^2|S_n] = S_n^2 + 2S_nE[A_n|S_n] + E[A_n^2|S_n] = S_n^2 + 0 + (1/4)1_{{S_n notin{0, 10}}}$$
        where $1_{{S_n notin{0, 10}}}$ is an indicator function that is 1 if $S_n notin {0,10}$ and is 0 else. So
        $$E[S_{n+1}^2] = E[S_n^2] + (1/4)P[S_n notin {0,10}]$$
        Subtracting 25 from both sides gives
        $$ Var(S_{n+1}) = Var(S_n) + (1/4)P[S_n notin {0,10}]$$
        and $Var(S_0)=0$ so
        $$ boxed{Var(S_n) = (1/4)sum_{i=0}^{n-1} P[S_i notin {0,10}] quad forall nin {1, 2, 3, ...} } $$
        Since $P[S_i notin {0,10}] = 1$ for $i in {0, 1, 2, 3, ..., 9}$ we have $$boxed{Var(S_1)=1/4, Var(S_2)=2/4, Var(S_3) = 3/4, ..., Var(S_{10})= 10/4}$$
        On the other hand:
        $$ Var(S_{11}) = 10/4 + (1/4)underbrace{(1-2(1/2)^{10})}_{P[S_{10}notin{0,10}]}$$
        In general, the variance increases as $nrightarrowinfty$ to approach a
        limiting value of $25$. It is possible to compute $P[S_i notin {0,10}]$ for all $i$ (for example, by taking powers of a transition probability matrix), but this calculation is more involved.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 23 at 19:51

























        answered Nov 23 at 19:40









        Michael

        13.2k11325




        13.2k11325






























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