Finding number of points in a bounded set when number of points in the unbounded set are known.

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Consider a random distribution of points in a Random 2D plane. I would like to find the number of points in a circle within this plane. Can anybody helps in solving the problem?

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That completely depends on the distribution of the points. In general, nothing can be said. For example, if the distribution of points is gaussian with mean $(0,0)$, then there will be much more points in the circle with center $(0,0)$ and radius $1$ than in the circle with center $(100000,100000)$ and radius $1$.