How to simulate the exponential law over an interval of the form [0,T]?

Question

How do you simulate an exponential random variable over an interval $[0, T]$ with $T > 0$?

Answer 1

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Let $X $ be an exponential r.v. of parameter $\lambda $ $$P (X<u|X <T)=\frac {P (X<min (u,T))}{P (X <T )} $$ So for $0\leq u\leq T$ $$P (X<u|X <T)=\frac {1-\exp (-\lambda u)}{1-\exp (-\lambda T)} $$

So if $U $ is an uniform on $ [0,1]$ then $$Y= -\frac {1}{\lambda }\ln\left (1-U (1-\exp(-\lambda T))\right) $$ is an exponential r.v. of parameter $\lambda $ conditionned to be on $[0,T] $