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# PROBABILITY ASSIGNMENT 2 1. Let A1 , A2 , ... be a sequence of events in . (a) Assume that {An }n is periodic (meaning that there is a nite number of...

Let (
, F, P) be a probability space, and let X :
! R, Y :
! R be random
variables. Define:
Z(!) = min{X(!), Y (!)}
T(!) = max{X(!), Y (!)}
Show that T,Z are random variables.
4. Let X, Y be random variables.
(a) Prove that X2 is a random variable.
(b) Prove that XY is a random variable.
Hint: think how you can express XY so that you’ll need only (a) and results
from class.
5. Let (
, F, P) be a probability space, where
PROBABILITY ASSIGNMENT 2 1. Let A 1 ,A 2 ,... be a sequence of events in Ω. (a) Assume that { A n } n is periodic (meaning that there is a ﬁnite number of sets B 1 ,B 2 ,...,B k such that A nk + j = B j for every j = 1 ,...,k and n = 0 , 1 ,... ). Find lim sup A n and lim inf A n . (b) Let n 1 ,n 2 ,... be an increasing sequence of natural numbers and consider the subsequence of sets { A n k } k . What can you say about the link between lim sup A n and lim sup A n k ? what about lim inf A n and lim inf A n k ? 2. Let I B : Ω R be the indicator function of the event B Ω. Show that if { A n } n is a sequence of subsets of Ω then I lim inf A n lim inf I A n and I lim sup A n lim sup I A n 3. Let (Ω ,F, P ) be a probability space, and let X : Ω R , Y : Ω R be random variables. Deﬁne: Z ( ω ) = min { X ( ω ) ,Y ( ω ) } T ( ω ) = max { X ( ω ) ,Y ( ω ) } Show that T,Z are random variables. 4. Let X,Y be random variables. (a) Prove that X 2 is a random variable. (b) Prove that XY is a random variable. Hint: think how you can express XY so that you’ll need only (a) and results from class. 5. Let (Ω , F , P ) be a probability space, where Ω = { ( x,y ) R 2 : x 2 + y 2 1 } , F the class of all Borel subsets of Ω, and P the probability measure on Ω deﬁned according to the Lebesgue measure. (a) What is the probability of the event A r = { ( x,y ) : p x 2 + y 2 r } ? (b) What is the probability that the angle between the line connecting a point and the origin, and the positive part of the x -axis, be smaller than α , α ( - π,π ]? 1

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