Solution
= Solution
{c}
For the rate-one <exponential distribution>, the <probability density function> is $e^{-t}$ for $t\ge0$. By <integration by parts>,
$$
\mathbb E T=\int_0^\infty te^{-t}\,dt=1,\qquad
\mathbb E(T^2)=\int_0^\infty t^2e^{-t}\,dt=2.
$$
Thus
$$
\boxed{\mathbb E T=1,\qquad\operatorname{Var}(T)=2-1^2=1.}
$$