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Convolute and geometrical probability spaces

Major M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2011

Vol. 16

255--263

Let X and Y be two independent random variables, either discrete or continuous. The question is "what is the probability distribution of Z = X + Y"? Clearly, the probability distribution of Z = X + Y is some combination of fX and fY which is called the convolution of fX and fY. It is denoted by ∗. We have fZ(t) = fX+Y(t) = fX(t)∗fY(t). In this paper it is shown how we can use geometrical probability spaces to find (without convolution) the distribution of random variable Z = X + Y.

Extension theorems in axiomatic theory of convexity

Kubiś W.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

89-96

We present a criterion for extending convexity preserving maps of convexity spaces. In a special case of convexity generated by a lattice structure this gives Sikorski's Extension Criterion for extending of maps of lattices. We also consider the class of convexity absolute extensors. It appears that complete Boolean algebras with a natural convexity belong to this class. In particular, we present an analogue of Tietze-Urysohn's Extension Theorem for maps of convexity spaces with values in a complete Boolean algebra.