We know that, for a 1D symmetrical random walk, ${\displaystyle p = 1/2, q = 1/2}$, with equal walk step length, after n steps, its probability distribution will be proportional to binomial coefficient:
${\displaystyle f(k) = {n \choose k}}$
or in the continuous limit, its probability distribution will be Gaussian-normal distribution.
My question is, which type of random walk can have "scaled" binominal coefficient distribution ? that is, its distribution is:
${\displaystyle g(k) = f(ak), a > 0}$
I tried different walk step length for "walk to the left" and "walk to the right", (asymmetrical random walk), I could not get above distribution.
Can anyone help ?
Thank you in advance.