1D random walk probability distribution I am way more physicist than mathematician and this question arises from experimental physics/engineering. The motivation is dealing with small amount of random discrete shifts between measured periods of digital signal:


*

*Suppose there is a $1D$ random walk with a possibility of remaining at the place, i.e. steps $\left(~-k,-k + 1,\ldots,\pm 0,\ldots,+k~\right)$ for
$k \in \mathbb{N}$ in each step.

*The probability is symmetric and equal for all $2k + 1$ possibilities $\left(~\mbox{i.e.}\ p=0.2\ \mbox{for}\ k = 2~\right)$.

*I need a formula for probability of reaching a particular discrete distance of $x$ after $n$ steps.


It must be a solved problem, I just didn't get lucky googling.
 A: Your problem is equivalent to toss a $2k+1$ facets die at each step, subtract $k+1$ and get the
result as the $\Delta x$ to move.
Equivalently you can toss a die, with $2k+1$ facets, numbered $0,\; \ldots ,\,r=2k$, and subtract $k$.
Let's take this model (it simplifies the treatment).
So, you are asking what is the probability that after $m$ tosses you'd get $s=X+mk$.
The total number of equi-probable events will be $(2k+1)^m$.
That is the number of integer points in a $m$-dimensional cube with side $0,\; \ldots ,\,r$.
The number of events leading to $s$ will be
$$
N_{\,b} (s,r,m) = \text{No}\text{. of solutions to}\;\left\{ \begin{gathered}
  0 \leqslant \text{integer  }x_{\,j}  \leqslant r \hfill \\
  x_{\,1}  + x_{\,2}  +  \cdots  + x_{\,m}  = s \hfill \\ 
\end{gathered}  \right.
$$
which geometrically is the number of integer points on the diagonal plane  summing to $s$, delimited to be within the cube, and
which is given by
$$
N_{\,b} (s,r,m)\quad \left| {\;0 \le {\rm integers}\;s,r,m} \right.\quad  = \sum\limits_{\left( {0\, \le } \right)\,j\,\left( { \le \,{s \over r}\, \le \,m} \right)} {\left( { - 1} \right)^{\,j} \left( \matrix{
  m \cr 
  j \cr}  \right)\left( \matrix{
  s + m - 1 - j\left( {r + 1} \right) \cr 
  s - j\left( {r + 1} \right) \cr}  \right)}
$$
Refer to Rolling dice problem for further considerations.
