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Neither (2) nor (3) are necessary. Note that, by definition, $E(T_1;T_1\lt T_2)$ is $$ \int_0^\infty uf_{T_1}(u)\int_u^\infty f_{T_2}(v)\mathrm dv\mathrm du=\int_0^\infty \lambda_1u\mathrm e^{-\lambda_1 u}P(T_2\gt u)\mathrm du. $$ Can you finish this?

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