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Math Help - Mean of a probability density function

  1. #1
    DannyBoy
    Guest

    Question 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 ?

    Thanks in advance!
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  2. #2
    MHF Contributor
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    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'.
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