Find the probability of encounter between two flies on the plane (similar to gambler's ruin problem).
$begingroup$
Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.
I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.
My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?
Any help will be much appreciated.
probability-theory stochastic-processes recurrence-relations
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
$endgroup$
add a comment |
$begingroup$
Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.
I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.
My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?
Any help will be much appreciated.
probability-theory stochastic-processes recurrence-relations
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
$endgroup$
1
$begingroup$
Try to write down an equation for the distance between the two flies...
$endgroup$
– Fabian
Nov 24 '18 at 12:34
add a comment |
$begingroup$
Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.
I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.
My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?
Any help will be much appreciated.
probability-theory stochastic-processes recurrence-relations
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
$endgroup$
Two flies sits on the plane. First one sits at $(0,0)$ and the second one at $(0,8)$, where first coordinate indicates time and second indicates location. Then, they start to move randomly and independently. First one goes $1$up with probability $frac13$ or $1$down with probability $frac23$. Second one goes $1$up with probability $frac34$ or $1$down with probability $frac14$. If a distance between them equals to $12$ - flies fly away. Find the probability of encounter and the average time to encounter or to fly out.
I know that I have to find a proper difference equation. It's a similar exercise to gambler's ruin, but now there are "$2$ people instead of $1$" and I have to find difference equation depending on distance.
My guess is: $psi_k=frac13psi_{k+1}+frac23psi_{k-1}$ for the first one and $psi_k=frac34psi_{k+1}+frac14psi_{k-1}$ for the second one, then I just compare them. Or do I calculate them independently with different boundary values?
Any help will be much appreciated.
probability-theory stochastic-processes recurrence-relations
probability-theory stochastic-processes recurrence-relations
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
asked Nov 24 '18 at 12:11
MacAbraMacAbra
16619
16619
1
$begingroup$
Try to write down an equation for the distance between the two flies...
$endgroup$
– Fabian
Nov 24 '18 at 12:34
add a comment |
1
$begingroup$
Try to write down an equation for the distance between the two flies...
$endgroup$
– Fabian
Nov 24 '18 at 12:34
1
1
$begingroup$
Try to write down an equation for the distance between the two flies...
$endgroup$
– Fabian
Nov 24 '18 at 12:34
$begingroup$
Try to write down an equation for the distance between the two flies...
$endgroup$
– Fabian
Nov 24 '18 at 12:34
add a comment |
1 Answer
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$begingroup$
Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?
Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
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add a comment |
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1 Answer
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$begingroup$
Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?
Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
$endgroup$
add a comment |
$begingroup$
Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?
Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
$endgroup$
add a comment |
$begingroup$
Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?
Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
$endgroup$
Construct a Markov chain $(X_n)_{n in mathbb{N}_0}$ that models the distance between the two flies. The chain has the possible states ${0,2,4,6,8,10,12}$. (We just say that once the chain has reached either $0$ or $12$, it stays there.) What are the transition probabilities?
Now let $p_i := P((X_n) text{ reaches } 0 text{ before } 12 | X_0 = i)$ for $i in {0,2,4,6,8,10,12}$. (You are looking for the value of $p_8$.) What are the values of $p_0$ and $p_{12}$? Do first step analysis to relate the other $p_i$'s to each other to get a system of linear equations that you can solve.
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
answered Nov 24 '18 at 17:12
Tki DenebTki Deneb
29210
29210
add a comment |
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$begingroup$
Try to write down an equation for the distance between the two flies...
$endgroup$
– Fabian
Nov 24 '18 at 12:34