Question

Suppose your experiment is rolling a die once. The sample space is Ω={1,2,…,6}. Consider the event A={ω∈Ω∣ω∈{1,3,5}}, which is the event that the number has shown up is odd. Recall from L2 the notation σ(A) which is the σ-algebra generated by A. a. Find σ(A). b. Consider the map X:Ω→R X(w) ={1 w ∊ { 1,2,3 } ,-1 otherwise.

​Is X is random variable on the measurable space (Ω,σ(A)) ? Mathematically explain your answer.

Answer 1

a. The sigma-algebra generated by event A, denoted σ(A), is the collection of all subsets of the sample space Ω that contain A and are closed under complementation and countable unions. In this case, A={1,3,5}, so σ(A) = {A, {}, {1,3,5}, {2,4,6}}. b. The map X:Ω→R is a random variable on the measurable space (Ω,σ(A)); X(w) ={1 if w is in {1,2,3}, -1 otherwise. X is a random variable on (Ω,σ(A)).

Step-by-step explanation:

a. The sigma-algebra generated by event A, denoted σ(A), is the collection of all subsets of the sample space Ω that contain A and are closed under complementation and countable unions. In this case, A={1,3,5}, so the subsets that contain A are A itself and the empty set {}. The complement of A is {2,4,6}. The countable unions of subsets containing A are {1,3,5}, {}, and {2,4,6}. Therefore, σ(A) = {A, {}, {1,3,5}, {2,4,6}}.

b. The map X:Ω→R is a random variable on the measurable space (Ω,σ(A)) if for every real number a, the set {w∈Ω:X(w)≤a} is an element of σ(A). In this case, X(w) ={1 if w is in {1,2,3}, -1 otherwise. Let a be a real number. If a≥1, then {w∈Ω:X(w)≤a} = Ω, which is an element of σ(A). If a<-1, then {w∈Ω:X(w)≤a} = {}, which is an element of σ(A). If -1≤a<1, then {w∈Ω:X(w)≤a} = {2,4,6}, which is an element of σ(A). Therefore, X is a random variable on the measurable space (Ω,σ(A)).