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Maxima voor de eenzijdige overschrijdingskans en voor de grootte van de gemiddelde overschrijding bij discontinue en continue, eentoppige verdelingen
Author(s) -
Goudriaan J.
Publication year - 1962
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1962.tb01069.x
Subject(s) - mathematics , limit (mathematics) , maxima , combinatorics , mathematical analysis , art , performance art , art history
Summary The well‐known inequality of Bienaymé‐Tschébyschef (for short B‐T), generalized by Camp and Meidell (for short C‐M) for continuous, unimodal distributions gives specific limits for total probabilities outside the ± to limits. In many cases however, especially in the field of industrial applications we are interested only in the probability of one tail of the distribution, which of course must be smaller than the limits given by the B‐T and C‐M formula. For these cases the maximum probability of surpassing the to limit on one side equals under B‐T conditions and under C‐M conditions instead of the two‐sided values of 1/t 2 and 47/9 · 1/t 2 respectively (cf e.g. Uspensky: “Introduction to mathematical probability”, 1937, p. 198) These results set upper limits for the value of Alternatively we may also set an upper limit for the integral which measures in terms of σ the average amount by which the limit x̄+ tσ is exceeded. This problem is also discussed and under C‐M conditions an upper limit is derived. Some practical applications of these results are considered.