Stochastic Calculus: [imath]\sigma[/imath]-algebra [imath]\mathcal{F}_2[/imath] (Toss a fair coin for 5 times. Describe the sample space Ω.)

[Win_odd Dhamnekar]

Toss a fair coin for 5 times. Describe the sample space [imath]\Omega.[/imath] We want to consider functions [imath]X : \omega \rightarrow \{-1,1\}.[/imath] Now describe the probability space [imath]( \Omega, \mathcal{F}_2, \mathbb{P}_2) [/imath] that arises if we want that only combinations of sets of the type [imath]A_i = \{ \omega \in \Omega :\text{ Number of heads} = i\}, i =\{ 0,1,2, 3,4,5 \}[/imath] should be events. How many [imath]\mathcal{F}_2 [/imath] measurable functions X with [imath]\mathbb{E}[X]=0 [/imath] are there?

My attempted answer:

The sets [imath]A_0 = \{T T T T\} ,A_1 = \{T T T H, T T H T, T H T T, H T T T \}, A_2 = \{T T H H, T H H T, H H T T, H T T H, T H T H, H T H T \}, A_3 = \{H H H T, H H T H, H T H H, T H H H\} A_4 = \{H H H H\}[/imath] defnes a partition of [imath]\Omega[/imath]

Now, what is [imath] \sigma[/imath]-algebra [imath]\mathcal{F}_2[/imath] generated by this partition of [imath]\Omega ?[/imath] and what is [imath]\mathbb{P}_2 ?[/imath]

Define [imath]\mathbb{P}_i= \mathbb{P}[A_i][/imath] ,then [imath]\mathbb{P}_0 =\frac{1}{16}, \mathbb{P}_1= \frac{4}{16}, \mathbb{P}_2= \frac{6}{16}, \mathbb{P}_3=\frac{4}{16}, \mathbb{P}_4=\frac{1}{16}[/imath]

For the set [imath]A_1[/imath], [imath]\sigma[/imath]-algebra [imath]\mathcal{F}[/imath] is [imath]\{\phi, \Omega, A_1, A^c\}[/imath]

Above all, How to answer this question?

 

[Win_odd Dhamnekar]

Toss a fair coin for 5 times. Describe the sample space [imath]\Omega.[/imath] We want to consider functions [imath]X : \omega \rightarrow \{-1,1\}.[/imath] Now describe the probability space [imath]( \Omega, \mathcal{F}_2, \mathbb{P}_2) [/imath] that arises if we want that only combinations of sets of the type [imath]A_i = \{ \omega \in \Omega :\text{ Number of heads} = i\}, i =\{ 0,1,2, 3,4,5 \}[/imath] should be events. How many [imath]\mathcal{F}_2 [/imath] measurable functions X with [imath]\mathbb{E}[X]=0 [/imath] are there?

My attempted answer:

The sets [imath]A_0 = \{T T T T\} ,A_1 = \{T T T H, T T H T, T H T T, H T T T \}, A_2 = \{T T H H, T H H T, H H T T, H T T H, T H T H, H T H T \}, A_3 = \{H H H T, H H T H, H T H H, T H H H\} A_4 = \{H H H H\}[/imath] defnes a partition of [imath]\Omega[/imath]

Now, what is [imath] \sigma[/imath]-algebra [imath]\mathcal{F}_2[/imath] generated by this partition of [imath]\Omega ?[/imath] and what is [imath]\mathbb{P}_2 ?[/imath]

Define [imath]\mathbb{P}_i= \mathbb{P}[A_i][/imath] ,then [imath]\mathbb{P}_0 =\frac{1}{16}, \mathbb{P}_1= \frac{4}{16}, \mathbb{P}_2= \frac{6}{16}, \mathbb{P}_3=\frac{4}{16}, \mathbb{P}_4=\frac{1}{16}[/imath]

For the set [imath]A_1[/imath], [imath]\sigma[/imath]-algebra [imath]\mathcal{F}[/imath] is [imath]\{\phi, \Omega, A_1, A^c\}[/imath]

Above all, How to answer this question?

ERRATA to my attempted answer :

[imath]A_5 =\{HHHHH\},[/imath]

[imath] A_4=\{HHHHT, HHHTH, HHTHH, HTHHH,THHHH\}, [/imath]

[imath]A_3=\{HHHTT, HHTTH, HTTHH, TTHHH, HTHTH, THTHH, HHTHT, HTHHT,THHTH,THHHT\}, [/imath]

[imath]A_2= \{TTTHH, TTHHT, THHTT, HHTTT, THTHT, HTHTT, TTHTH, THTTH, HTTHT, HTTTH\},[/imath]

[imath] A_1=\{TTTTH, TTTHT, TTHTT,THTTT,HTTTT\},A_0 =\{TTTTT\}[/imath]

Define [imath]\mathbb{P}_i = \mathbb{P}[A_i][/imath], then [imath] \mathbb{P}_0 =\displaystyle\frac{1}{32}, \mathbb{P}_1= \displaystyle\frac{5}{32}, \mathbb{P}_2 = \displaystyle\frac{10}{32}, \mathbb{P}_3 = \displaystyle\frac{10}{32}, \mathbb{P}_4= \displaystyle\frac{5}{32}, \mathbb{P}_5= \displaystyle\frac{1}{32}[/imath]

For the set [imath]A_1[/imath], [imath]\sigma[/imath]-algebra [imath]\mathcal{F}[/imath] is [imath]\{\phi, \Omega, A_1, A^c\}[/imath]

What are [imath]\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3, \mathcal{F}_4, \mathcal{F}_5,[/imath] measurable sets in this problem?

How to generate [imath]\sigma[/imath]- algebra [imath]\mathcal{F}_0, \mathcal{F}_1, \mathcal{F}_2, \mathcal{F}_3, \mathcal{F}_4, \mathcal{F}_5 ?[/imath]

 

[Win_odd Dhamnekar]

My answer to this question:
There are [imath]{32}\choose{8}[/imath] =10518300 possible [imath]\mathcal{F}_2[/imath] measurable functions [imath]X :\Omega \rightarrow \{-1,1\} [/imath] such that [imath]\mathbb{E}[X]=0 [/imath]