A ball is residing at the origin of an n-dimensional grid. In a specific time period, it moves along some of the axes, in the positive direction with probability p. For every direction, it remains stationary along that axis with probability 1-p. How will you describe the distribution of the distance traversed by the ball? Even if the question is ambiguous what are the possibilities from the below list?
- Chi-squared
- Gaussian
- Poisson
- Binomial
Let's say we are interested in the distribution of the distance traversed by the ball after $K$ periods. In the $j^{th}$ dimension, the distance traveled by the ball can be modelled as a binomial RV: $X_j(K) \sim B(K,p)$. The total distance traveled by the ball after $K$ periods is given by:
$X(K) = \sqrt{X_1(K)^2 + X_2(K)^2 + \;...\; + X_n(K)^2}$.
I don't know if a closed-form expression for such a function of $n$ binomial RVs can be derived. However, under suitable assumptions, one may approximate each binomial RV $X_j(K)$ with a normal RV $N(Kp, Kp(1-p))$. Under this approximation, the distance $X(K)$ will follow a non-central chi-squared distribution with $n$ degrees of freedom (https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution).