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

, 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

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