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Suppose the future lifetime of someone aged $20$ denoted by $T_{20}$ is subject to the force of mortality $\mu_x = \frac{1}{100-x}$ for $x< 100$. What is $\text{Var}[\min(T_{20},50)]$?

So we have: $E[\min(T_x,50)|T_{20} > 50] = 50$ $\text{Var}[\min(T_{20},50)|T_{20} > 50] = ?$ $E[\min(T_x,50)|T_{20} \leq 50] = 25$ $\text{Var}[\min(T_{20},50)|T_{20} \leq 50] = 50^{2}/12$

What is the second line?

Also I know that we use the "expectation-variance" formula to calculate $\text{Var}[\min(T_{20},50)]$.

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Hint: The random variable $\min(T_{20},50)|T_{20} > 50)$ doesn't vary much!

If you then want to use a formula, call the above random variable $Y$. We want $E((Y-50)^2)$. How did you decide earlier that $E(Y)=50$?