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### how to generate random number of a random variable's residual life

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```I have a question about random number generation.

Suppose that X is an Erlang distributed random variable with
probability density function(pdf) f_X(t) and cumulative distribution
function (CDF) F_X(t). Let Y as the residual life of X with pdf f_Y(t).

Referring to the residual life theorem in the book L. Kleinrock,
Queueing Systems, then,
f_Y(t) = (1/ (mean of X)) [1-F_X(t)].

It is easy to generate Erlang random number samples for X. In such
case, how to generate the random number samples for Y? is it possible
to derive the random number samples for Y from the samples for X?

ZHANG Yan

```
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Reply buaanupt (78) 5/14/2006 3:57:45 PM

```<buaanupt@sina.com> wrote in message
>I have a question about random number generation.
>
>
> Suppose that X is an Erlang distributed random variable with
> probability density function(pdf) f_X(t) and cumulative distribution
> function (CDF) F_X(t). Let Y as the residual life of X with pdf f_Y(t).
>
> Referring to the residual life theorem in the book L. Kleinrock,
> Queueing Systems, then,
> f_Y(t) = (1/ (mean of X)) [1-F_X(t)].
>
>
> It is easy to generate Erlang random number samples for X. In such
> case, how to generate the random number samples for Y? is it possible
> to derive the random number samples for Y from the samples for X?
>
>

If U is a variate with uniform distribtion on (0,1) and F(x) is a strictly
increase cdf, then

X = F^(-1)(U) is a variate with distribution F(x)

hence you can generate a uniformly distributed random number and use the
inverse of F to get a random number that is distributed like F.

you might have to approximate F^(-1) and it could take a long time but it
should do the job.

Jon

```
 0

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5/24/2013 6:51:28 AM