Question

Let Ω={−2,−1,0,1}. Let X(ω)=ω2 be a random variable defined on Ω. Find the smallest sigma-algebra generated by X.

Answer 1

The smallest sigma-algebra generated by the random variable X(ω) = ω^2 is the sigma-algebra generated by the events {−2,−1}, {0} and {1} in Ω.

Step-by-step explanation:

The smallest sigma-algebra generated by a random variable X is generated by the preimages of the sets in the range of X. In this case, the range of X is {0, 1, 4}, since X(ω) = ω^2. Therefore, the smallest sigma-algebra generated by X is the collection of sets whose preimages under X are either empty, Ω, or one of the sets {−2,−1}, {0} or {1}, which is the sigma-algebra generated by the events {−2,−1}, {0} and {1} in Ω.