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A highly accurate and efficient method to compute the expected values of thecount, sum, and squared norm of the sum of the centre vectors of a randommaximal sized collection of non-overlapping unit diameter disks touching afixed unit-diameter disk is presented. This extends earlier work on R\'enyi'sparking problem [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 3 (1-2),1958, pp. 109-127]. Underlying the method is a splitting of the the problemconditional on the value of the first disk. This splitting is proven and thenused to derive integral equations for the expectations. These equations take alower block triangular form. They are solved using substitution andapproximation of the integrals to very high accuracy using a polynomialapproximation within the blocks.

Computing expected moments of the R\'enyi parking problem on the circle