= Solution
For <quota share reinsurance> the insurer retains the same fraction of each claim, so
$$
\boxed{g(x)=\alpha x.}
$$
The annual retained loss is consequently $T_I=\alpha T$. Each transformed risk severity has <probability density function> $f_i(y/\alpha)/\alpha$ for $y>0$. The mixed transformed severity, with the same weights $\lambda_i/(\lambda_1+\lambda_2)$, still gives a <compound Poisson distribution>. Scaling the <expected value> and <variance> gives
$$
\boxed{\mathbb ET_I=\alpha\sum_{i=1}^2\lambda_i\mu_i,\qquad
\operatorname{Var}(T_I)=\alpha^2\sum_{i=1}^2\lambda_i(\sigma_i^2+\mu_i^2).}
$$
This agrees with the general retained-claim formulas because $\mathbb E g(X_i)=\alpha\mu_i$ and $\mathbb E[g(X_i)^2]=\alpha^2(\sigma_i^2+\mu_i^2)$.
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