The probability of a random walker hitting a barrier for the first time
I am grabbling with the following problem and the final condition gets me. To me it seems that I must do an awful lot of accounting to keep track of instances where the random variable has already accumulated more than $S$, before depreciating and increasing to $S$ once more, and so on, and so forth. Is there a simple trick to this accounting? If so, I would be grateful to hear what is:
A random variable increments by +1 every $t$ period with probability $p$, and decrements by -1 every $t$ period with probability $1 – p$. What is the probability of the random variable having accumulated to exactly $S$ after an arbitrary period of time, given that it has never been higher than $S$? In other words, what is the probability of the random variable reaching $S$ for the first time after an arbitrary number of periods?
random-variables random-walk
asked Nov 18 at 22:53
user120911
233110
add a comment |
I am grabbling with the following problem and the final condition gets me. To me it seems that I must do an awful lot of accounting to keep track of instances where the random variable has already accumulated more than $S$, before depreciating and increasing to $S$ once more, and so on, and so forth. Is there a simple trick to this accounting? If so, I would be grateful to hear what is:
A random variable increments by +1 every $t$ period with probability $p$, and decrements by -1 every $t$ period with probability $1 – p$. What is the probability of the random variable having accumulated to exactly $S$ after an arbitrary period of time, given that it has never been higher than $S$? In other words, what is the probability of the random variable reaching $S$ for the first time after an arbitrary number of periods?
random-variables random-walk
asked Nov 18 at 22:53
user120911
233110
See mathoverflow.net/questions/124242/… or math.stackexchange.com/questions/64919/…
– Stockfish
Nov 18 at 23:09
add a comment |
I am grabbling with the following problem and the final condition gets me. To me it seems that I must do an awful lot of accounting to keep track of instances where the random variable has already accumulated more than $S$, before depreciating and increasing to $S$ once more, and so on, and so forth. Is there a simple trick to this accounting? If so, I would be grateful to hear what is:
A random variable increments by +1 every $t$ period with probability $p$, and decrements by -1 every $t$ period with probability $1 – p$. What is the probability of the random variable having accumulated to exactly $S$ after an arbitrary period of time, given that it has never been higher than $S$? In other words, what is the probability of the random variable reaching $S$ for the first time after an arbitrary number of periods?
random-variables random-walk
asked Nov 18 at 22:53
user120911
233110
I am grabbling with the following problem and the final condition gets me. To me it seems that I must do an awful lot of accounting to keep track of instances where the random variable has already accumulated more than $S$, before depreciating and increasing to $S$ once more, and so on, and so forth. Is there a simple trick to this accounting? If so, I would be grateful to hear what is:
A random variable increments by +1 every $t$ period with probability $p$, and decrements by -1 every $t$ period with probability $1 – p$. What is the probability of the random variable having accumulated to exactly $S$ after an arbitrary period of time, given that it has never been higher than $S$? In other words, what is the probability of the random variable reaching $S$ for the first time after an arbitrary number of periods?
random-variables random-walk
random-variables random-walk
asked Nov 18 at 22:53
user120911
233110
asked Nov 18 at 22:53
user120911
233110
edited Nov 18 at 23:01
asked Nov 18 at 22:53
user120911
233110
asked Nov 18 at 22:53
user120911
233110
asked Nov 18 at 22:53
user120911
233110
233110
See mathoverflow.net/questions/124242/… or math.stackexchange.com/questions/64919/…
– Stockfish
Nov 18 at 23:09
add a comment |
See mathoverflow.net/questions/124242/… or math.stackexchange.com/questions/64919/…
– Stockfish
Nov 18 at 23:09
See mathoverflow.net/questions/124242/… or math.stackexchange.com/questions/64919/…
– Stockfish
Nov 18 at 23:09
See mathoverflow.net/questions/124242/… or math.stackexchange.com/questions/64919/…
– Stockfish
Nov 18 at 23:09
add a comment |
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See mathoverflow.net/questions/124242/… or math.stackexchange.com/questions/64919/…
– Stockfish
Nov 18 at 23:09