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Let X0, X1, X2, ... ) random variables, that is, with common probability distribution and let N — min { k: Xk > X0 } . One can give the following interpre­ tation of N. Let a person, say Paul, play a game in which the value of prize is Probability Theory 34 random (it may be, for instance, a lottery) and let X0 be the size of the prize thus obtained. His friend Peter plays the same game several times till the size of his prize in a particular game exceeds Xg. If X 15 X2, ... are the sizes of Peter's prizes in the respective games, then N is the number of games to be played by Peter until the desired result is achieved.

J . a 5. 1. v. defined on that space. Definition 1. Expectation E X of a random variable X is the number E I : = E ^ ) P H (1) provided the series on the right-hand side of (1) is convergent. A n extension of Definition 1. Let £+:= E X(u,)pH, £-:= w:X(w)>0 E *(«)pH'- w:X(w)<0 If £ + < oo and | £ _ | < oo, then 'EX is defined as before. If £ + < oo and £ _ = — oo, then E X := —oo. If £ + = +do and | E - I < °°> then E X := +oo. If Y1+ = + ° ° an< ^ E- = —°°i then E X does not exist. Lemma 1.

In other words, E { Ic, Y \ (5 } = IC}E { Y | © } and E { x y | 0 } = E^-fc j E{r|®} = E { r | 0 } E * j / C j . The preceding relation together with (11) gives (16). cjr (17) r where j 1 : j 2 , . . is an arbitrary sequence of distinct integers. Then Ic = YjTIcir and E { / c F | © } = / c E { F | <&}. v. E { Y | <5 } and, therefore, E F = E [ E { F | 0}]. (19) Relations (18) and (19) allow us to conclude that under partition (17) E { Y; C } = E { YIC } = E [ E { YIC |

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