I have a problem which I have modelled as a set of IID Bernouilli variables, each of which has a location along a line (that is, each variable $\displaystyle X_i$ can be said to have a position $\displaystyle d_i$).

For a given data point (that is, for $\displaystyle \mathbf X = \mathbf x$), I am able to calculate the probability of observing this number of the variables in the 'on' state (or more). That is, $\displaystyle P(\sum_i X_i \ge \sum_i x_i)$, as this is given by the binomial distribution.

However, I would like to calculate,given the number of observed points, some probability mass function for the positions of these points along the line. Essentially I'm trying to find out whether or not they are 'clustered'. One way I thought of for doing this would be to calculate the variance of an observation (given by $\displaystyle \sum_i x_i d_i^2 - (\sum_i x_i d_i)^2$) and then find the probability that the variance be equal to or lower than this value. However, I have not been able to find a closed-form p.m.f. for the variance in position.

Does anyone know how I should go about this, or have another suggestion about how I could represent the probability that a given observation is randomly distributed in space?