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convert inline expression to own-line expression, for readability
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edit approved Aug 19, 2017 at 22:08
Dan Stowell
edit approved Aug 19, 2017 at 22:08
Dan Stowell
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The explicit formula is: $P[N_m=n]=(m/n)P[S_n=m]$, where $P[N_m=n]$ is the probability the position $m$ is hit after exactly $n$ steps, $S_n = X_1+X_2+\dots X_n$ and $P[S_n=m]$ is the probablity after $n$ steps the path to be at the position $m$. This last is well-known and is given by $P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}$. This

$$P[S_n=m]=\frac{n!}{[(n+m)/2]![(n-m)/2]!}p^{(n+m)/2}q^{(n-m)/2}.$$

This is true when $n$ and $m$ have the same parity, else the probability is zero. Additionally $m$ is positive and $n$ is greater than or equal to $m$. A similar expression can be found for negative $m$.

The explicit formula is: $P[N_m=n]=(m/n)P[S_n=m]$, where $P[N_m=n]$ is the probability the position $m$ is hit after exactly $n$ steps, $S_n = X_1+X_2+\dots X_n$ and $P[S_n=m]$ is the probablity after $n$ steps the path to be at the position $m$. This last is well-known and is given by $P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}$. This is true when $n$ and $m$ have the same parity, else the probability is zero. Additionally $m$ is positive and $n$ is greater than or equal to $m$. A similar expression can be found for negative $m$.

The explicit formula is: $P[N_m=n]=(m/n)P[S_n=m]$, where $P[N_m=n]$ is the probability the position $m$ is hit after exactly $n$ steps, $S_n = X_1+X_2+\dots X_n$ and $P[S_n=m]$ is the probablity after $n$ steps the path to be at the position $m$. This last is well-known and is given by

$$P[S_n=m]=\frac{n!}{[(n+m)/2]![(n-m)/2]!}p^{(n+m)/2}q^{(n-m)/2}.$$

This is true when $n$ and $m$ have the same parity, else the probability is zero. Additionally $m$ is positive and $n$ is greater than or equal to $m$. A similar expression can be found for negative $m$.

Added LaTeX markup.
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Stefan Kohl ♦
Stefan Kohl ♦
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The explicit formula is: P[N_m=n]=(m/n)P[S_n=m]$P[N_m=n]=(m/n)P[S_n=m]$, where P[N_m=n]$P[N_m=n]$ is the probability probability the position m$m$ is hittedhit after exactly n$n$ steps, S_n=X_1+X_2+..X_n $S_n = X_1+X_2+\dots X_n$ and P[S_n=m]$P[S_n=m]$ is the probablity after n$n$ steps the the path to be at the position m$m$. This last is well known-known and is given by P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2} by $P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}$. This This is true when n$n$ and m$m$ have the same parity, else the probability is is zero. Additionally m$m$ is positive and n$n$ is greater than or equal to m$m$. A A similar eapressionexpression can be found for negative m$m$.

The explicit formula is: P[N_m=n]=(m/n)P[S_n=m] where P[N_m=n] is the probability the position m is hitted after exactly n steps, S_n=X_1+X_2+..X_n and P[S_n=m] is the probablity after n steps the path to be at the position m. This last is well known and is given by P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}. This is true when n and m have the same parity else the probability is zero. Additionally m is positive and n is greater or equal to m. A similar eapression can be found for negative m.

The explicit formula is: $P[N_m=n]=(m/n)P[S_n=m]$, where $P[N_m=n]$ is the probability the position $m$ is hit after exactly $n$ steps, $S_n = X_1+X_2+\dots X_n$ and $P[S_n=m]$ is the probablity after $n$ steps the path to be at the position $m$. This last is well-known and is given by $P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}$. This is true when $n$ and $m$ have the same parity, else the probability is zero. Additionally $m$ is positive and $n$ is greater than or equal to $m$. A similar expression can be found for negative $m$.

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Nicos Hiotelis
Nicos Hiotelis
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The explicit formula is: P[N_m=n]=(m/n)P[S_n=m] where P[N_m=n] is the probability the position m is hitted after exactly n steps, S_n=X_1+X_2+..X_n and P[S_n=m] is the probablity after n steps the path to be at the position m. This last is well known and is given by P[S_n=m]=(n!/([(n+m)/2]![(n-m)/2]!))p^{(n+m)/2}q^{(n-m)/2}. This is true when n and m have the same parity else the probability is zero. Additionally m is positive and n is greater or equal to m. A similar eapression can be found for negative m.