# Expected value of $\frac{\overline X}{1-\overline X}$ when $X_i$'s are i.i.d $\mathsf{Beta}(\theta,1)$

I am trying to determine $$E\left[\frac{\overline X}{1-\overline X}\right]$$, where distribution of $$X_1,\ldots,X_n$$ is

$$f(x;\theta)=\theta x^{\theta−1}\quad,\, 0 < x < 1\,,\, \theta > 0$$

When I try it by definition of $$E[X]$$ how do I integrate $$\frac{\overline X}{1-\overline X}$$?

• Please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Mar 18 '16 at 20:49
• Are you only wondering how to integrate $\bar X / (1- \bar X)$? If so, this may be better suited for the Mathematics SE site. – gung - Reinstate Monica Mar 18 '16 at 20:50
• First try and figure out the distribution of $\bar{X}$. – Greenparker Mar 18 '16 at 20:51
• Also, specify what is the support of the distribution, and the possible values of $\theta$? Is it all $\mathbb{R}$? – Greenparker Mar 18 '16 at 20:53
• yes really the question wants me to find the bias of this estimator which is the MME for this specific distribution. But without this exected value i have no way to find the bias. Don't use the given distribution? find $\bar{X}$ pdf first? – qqq2 Mar 18 '16 at 20:54

1. $\bar{X} = 1/n \sum X$, call $Y = \sum X$. $Y$ has a known distribution. What is it?
2. $\bar{X} / (1-\bar{X}) = Y / (n - Y)$
3. $E[Y/(n-Y)] = \int Y/(n-Y) dY$
• i noticed in my book that it used notations like E sub $\theta$ ($\theta$ hat) why do they have this E sub $\theta$? this is for the bias formula – qqq2 Mar 18 '16 at 22:26
• let $\theta$=A then i shud read this as $\mathbb{bias}_A[A hat]$ = $\mathbb{E}_A[A hat]$ - A where $\mathbb{E}_A[A hat]$=$\int (Ahat) f(x) dx$ using the original f(x) since sub A implies with resp to my original $\theta$? – qqq2 Mar 18 '16 at 23:38