# A problem of inversion method for sampling

I am reading the book Stochastic Simulation, Algorithms and Analysis by Asmussen. On page 39, when talking about inversion method for sampling a distribution, he gave an example:

an r.v. distributed as the overshoot X − a given X > a can be generated as $F^{-1} (U (1-F (a)) − a$,

where $U$ is a r.v. uniform in $[0,1]$.

But I think the distribution is $F^{-1} (U (1-F (a)) + F(a)) - a$. I was wondering which is correct and why?

Thanks and regards!

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Let $Y=X$ with support on $[a,\infty)$. If $f_X$ is the p.d.f of $X$ and $f_Y$ is the pd.f. of $Y$ then

$f_Y(y) = \frac{f_X(y)}{1-F(a)}$ for $y>a$

Hence

$F_Y(y) = \frac{F_X(y)-F_X(a)}{1-F_X(a)}$

If $U=F_Y(Y)$ then as usual we have $U\sim Uni[0,1]$.

That is

$U=\frac{F_X(Y)-F_X(a)}{1-F_X(a)}$

Rearranging gives us

$Y=F_X^{-1}(U(1-F_X(a))+F_X(a))$

Thus the over shoot r.v. $X-a$ for $X>a$ can be generated as

$F_X^{-1}(U(1-F_X(a))+F_X(a))-a$

I agree with you.

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