Probability

New PDF release: Analytical Methods in Probability Theory: Proceedings

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By Dogeu D. (ed.), Lucaks E. (ed.), Rohatgi V.K. (ed.)

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This can be justified (providing that the series converges) as a regularity property, thinking that T can be approximated with random numbers with a finite but arbitrarily large number of values. We then get +∞ +∞ iP(T = i) = P(T ) = i=1 +∞ i(1 − p)i−1 p = p i=1 i(1 − p)i−1 = i=1 p 1 = , p2 p where we used that for |x| < 1 +∞ +∞ i x i−1 = i=1 i=1 d i d [x ] = dx dx +∞ xi = i=0 d dx 1 1−x = 1 . (1 − x)2 The geometric distribution is said to be “memoryless”. e. the conditional probability of no success up and including the (m +n)th trial given that there was no success up and including the nth trial is equal to the probability of no success up and including the m trial: everything starts from scratch.

Limit from the left: lim− F(y) = P(X < x); 3. 4. lim F(x) = 1; y→x x→ + ∞ lim F(x) = 0. ’s these extra properties will be satisfied, even if it is possible to consider cases where they do not hold true. It follows from 1 and 2 that P(X = x0 ) = P((X ≤ x0 ) − (X < x0 )) = F(x0 ) − F(x0− ) where F(x0− ) denotes lim x→x0− F(x). This limit always exists as F(x) is bounded non-decreasing. 1 (Discrete case) In the case of a random number X with discrete distribution I (X ) = {x1 , x2 , . } one has: P(X = xi ).

It follows that non-correlation does not imply in general stochastic independence. If m = n = 2, then there is just one solution so that non-correlation and stochastic independence coincide. This is the case of events: two events are non-correlated if and only if they are stochastically independent. In Sect. 11 we have shown that stochastic independence implies non-correlation and that in fact it implies non-correlation of any two functions of the random numbers. 14 Generating Function Let X be a random number with discrete distribution on a subset of N.

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Analytical Methods in Probability Theory: Proceedings by Dogeu D. (ed.), Lucaks E. (ed.), Rohatgi V.K. (ed.)


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