This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i (v_1P_i)^{\frac{\delta_1}{2}}}+ \pi \lambda_i \delta_2 v_1^{\delta_2-1}P_i^{\delta_2}(1-e^{-\beta_i (v_1P_i)^{\frac{\delta_2}{2}}})\right) $$
$$ f(v_2|v_1)=\left(\sum_{i=1}^K \pi \lambda_i \delta_1v_2^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i ((v_2-v_1)P_i)^{\frac{\delta_1}{2}}} + \pi \lambda_i \delta_2 v_2^{\delta_2-1}P_i^{\delta_2}(1-e^{-\beta_i ((v_2-v_1)P_i)^{\frac{\delta_2}{2}}})\right)$$ where $ P_i,\beta_i,\lambda_i,\delta_1,\delta_2,K $ are all constants $\forall i$. I have to write their product $$f(v_1)f(v_2|v_1)$$ in nice closed form. I have done the following till now, $$\left(v_1^{\delta_1-1}v_2^{\delta_1-1}\left(\sum_{i=1}^K \pi \lambda_i \delta_1 P_i^{\delta_1}\right)^2 e^{-\beta_i (v_2P_i)^{\frac{\delta_1}{2}}} + v_1^{\delta_2-1}v_2^{\delta_2-1}\left(\sum_{i=1}^K \pi \lambda_i \delta_2 P_i^{\delta_2}\right) ^2(1-e^{-\beta_i (v_2P_i)^{\frac{\delta_2}{2}}})\right)$$
but I am still missing two terms within the product which are the cross terms - should the cross terms that be inside $\sum_{i,j}$, can anyone help me write it nice closed form