# Thread: Mean of a probability density function

1. ## Mean of a probability density function

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

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

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

2. 1) Isn't the denominator of the infinite limit just unity?
2) This is why we have Rules of Thumb. Here are a couple:

Use the t-distribution until the sample size is "big enough".

The Normal is a good approximation of Binomial if n is big enough and p is close to 1/2.

3) I'm not convinced that the mean is a good measure of t being big enough. Outliers are bad!

My views. I welcome others'.