Probability of X ∈ [0, 1/2) for X(ω) = ω^2

Probability Calculation

Let Ω = [0, 1] with Borel sets and Lebesgue measure. We are given X(ω) = ω^2. We need to find P(X ∈ [0, 1/2)).

Since X(ω) = ω^2 is a continuous function, we have to find the probability of the set [0, 1/2) in the image of X.

First, we find the image of X. For any y in [0, 1], there exists a unique x in [0, 1] such that X(x) = y^(1/2). Therefore, the image of X is [0, 1].

Next, we find the inverse image of [0, 1/2). That is, we need to find the set of all points in [0, 1] whose image under X is in [0, 1/2).

Let y be a point in [0, 1/2). Then we need to solve the equation X(x) = y, which gives us x = y^(1/2). Therefore, the inverse image of [0, 1/2) under X is [0, 1/4).

Finally, we find the probability of the inverse image of [0, 1/2) under X.

P(X ∈ [0, 1/2)) = P([0, 1/4)) = Leb([0, 1/4)) = 1/4.

Therefore, P(X ∈ [0, 1/2)) = 1/4.