Given the closed interval with , let one-dimensional "cars" of unit length be parked randomly on the interval. The mean number of cars which can fit (without overlapping!) satisfies
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The mean density of the cars for large is
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(OEIS A050996). While the inner integral can be done analytically,
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where is the Euler-Mascheroni constant and is the incomplete gamma function, it is not known how to do the outer one
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(8)
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(9)
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where is the exponential integral. The slowly converging series expansion for the integrand is given by
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In addition,
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for all (Rényi 1958), which was strengthened by Dvoretzky and Robbins (1964) to
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Dvoretzky and Robbins (1964) also proved that
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Let be the variance of the number of cars, then Dvoretzky and Robbins (1964) and Mannion (1964) showed that
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(OEIS A086245), where
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and the numerical value is due to Blaisdell and Solomon (1970). Dvoretzky and Robbins (1964) also proved that
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and that
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Palasti (1960) conjectured that in two dimensions,
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but this has not yet been proven or disproven (Finch 2003).