Let = [0, 1]. Define X : R by X(w) = w, 0 w 1/2 w 1/2, 1/2

Let Ω = [0, 1]. Define X : Ω → R by X(w) =



w, 0 ≤ w ≤ 1/2 w − 1/2, 1/2 ≤ w ≤ 1 .

For any interval I ⊆ [0, 1], let P(I) =



I 2xdx. Determine the distribution function of X and use this to find P(X > 1/2), P(1/4 < X < 1/2), P(X < 1/2/X > 1/4).