Probability theory, probability space, statistics

[Sarina3003]

Homework Statement

Homework Equations

All needed are in the picture above (i hope so)

The Attempt at a Solution

to me it is extremely difficult because it is so complicated with many notations. Also, I actually don't know how to read the question properly to answer it
Is E(beta) is the event where we get even number of elements Beta ? I hope i get this point right. Otherwise, i have no idea what E means here.
One more point i can say is that the given pmf is Geometric distribution. My problem is i do not know how to start the question

Please help me to clarify it and answer any part that you can. You don't have to answer all of them or be so specific but please give me as much explanation as possible until you think that you give me enough clues to proceed with the subsequent questions.

 

[Ray Vickson]

Homework Statement

View attachment 223980

Homework Equations

All needed are in the picture above (i hope so)

The Attempt at a Solution

to me it is extremely difficult because it is so complicated with many notations. Also, I actually don't know how to read the question properly to answer it
Is E(beta) is the event where we get even number of elements Beta ? I hope i get this point right. Otherwise, i have no idea what E means here.
One more point i can say is that the given pmf is Geometric distribution. My problem is i do not know how to start the question

Please help me to clarify it and answer any part that you can. You don't have to answer all of them or be so specific but please give me as much explanation as possible until you think that you give me enough clues to proceed with the subsequent questions.

The question tells you exactly what $E_\beta$ means, and it even gives you an example. It says "For example, with $\beta = (0,1,1,0),$ $E_\beta$ consists of all sequences of the form $(0,1,1,0,\alpha_5, \alpha_6, \ldots )$." In other words, for any finite string $\beta$ of 0s and 1s, $E_\beta$ consists of all those infinite sequences of 0s and 1s that start with the string $\beta$.