I have a probability density function (PDF), $\displaystyle \vartheta(r)$, where $\displaystyle r\in[0,\infty]$. So the mean of the PDF ($\displaystyle \bar{r}$) is calculated by $\displaystyle \int_0^\infty r\vartheta(r)dr$.

One can say that $\displaystyle \lim_{t\to\infty}\frac{\int_0^t r\vartheta(r)dr}{\int_0^t \vartheta(r)dr}=\bar{r}$.

But is it correct to state that $\displaystyle \frac{\int_0^t r\vartheta(r)dr}{\int_0^t \vartheta(r)dr}\approx\bar{r}$ when $\displaystyle t\gg\bar r$ ?

Thanks in advance!