Let X ~ exp(1), Y=e^{-X }and consider the simple linear model $\displaystyle Y = \alpha + \beta X + \gamma X^2 + W$, where$\displaystyle E(W)=0=\rho (X,W) = \rho(X^2,W)$.

Demonstrate that 1, X, X^{2}are linearly independent in L_{2}.

It also gives a hint: exp(1) = G(1), (gamma distribution with p=1)

I'm not sure how to show linear independence in L_{2}, I'm not even quite sure what L_{2}means exactly. Would showing Cov(1,X) = Cov(X,X^2) = Cov(1,X^2) = 0 be enough for linear independence? I'm also no sure how to use the hint..