Let (X₁, X₂) be random variables. Determine whether (X₁, X₂) are independent in each of the following examples:

f(x₁, x₂) = 12x₁x₂(1 x₂) for x₁ [0, 1], x₂ [0, 1]; 0 elsewhere.

f(x₁, x₂) = e ^x₁ e ^x₂ for x₁ > 0, x₂ > 0; 0 elsewhere.

f(x₁, x₂) = 1/ for x₁² + x₂² 1; 0 elsewhere.

M_XY (t₁, t₂) = (1/(1t₁)(1t₂)) , t₁ < 1, t₂ < 1.

M_XY (t₁, t₂) = exp(t₁ + t₂ + 0.5(t₁² + t₂² + 0.5t₁t₂)).

Hint: Think about factorization theorems. No need for complicated calculations!