Example 1 The marginal pdfs and expectations of X and Y are fx(X)=3xdy=3x2,0≤x≤1， EW=x3x2=居x-子 )=3xdx=)0sys1 m=a-9w=告--君 m=w3d-,=品Example 1 • The marginal pdfs and expectations of X and Y are 𝑓𝑋 𝑋 = න0𝑥 3𝑥𝑑𝑦 = 3 𝑥 2 , 0 ≤ 𝑥 ≤ 1 , E 𝑋 = න0 1 𝑥 ⋅ 3 𝑥 2𝑑𝑥 = 34 𝑥 4 01 = 34 , 𝑓𝑌 𝑦 = න𝑦 1 3𝑥𝑑𝑥 = 32 𝑥 2 𝑦1 = 32 1 − 𝑦 2 , 0 ≤ 𝑦 ≤ 1 , E 𝑌 = න0 1 𝑦 ⋅ 32 1 − 𝑦 2 𝑑𝑦 = 32 𝑦 22 − 𝑦 44 01 = 38 , E 𝑋𝑌 = න0 1 න0 𝑥 𝑥𝑦 ⋅ 3𝑥𝑑𝑦𝑑𝑥 = 32 𝑥 55 01 = 3 10 4