Consider a compound Poisson process ST=∑i=1NTXiS_T = \sum_{i=1}^{N_T} X_iST=∑i=1NTXi, where NT∼Poisson(λT)N_T \sim \text{Poisson}(\lambda T)NT∼Poisson(λT) and Xi∼Exp(β)X_i \sim \text{Exp}(\beta)Xi∼Exp(β). What is the variance Var(ST)Var(S_T)Var(ST)?
λT(1β2)\lambda T (\frac{1}{\beta^2})λT(β21)
λT(2β2)\lambda T (\frac{2}{\beta^2})λT(β22)
λT(1β+1β2)\lambda T (\frac{1}{\beta} + \frac{1}{\beta^2})λT(β1+β21)
λTβ2\frac{\lambda T}{\beta^2}β2λT