Math Help - expected value of division

1. expected value of division

Hi!

The task is to determine if estimate of the parameter is unbiased. In order to do that, i must find expected value of the parameter, given distribution of each observation which are iid.

I have come to this $E(\frac{1}{\sum_{i=1}^N X_i})$, where $EX_i$ is given. I think in general $E(\frac{1}{\sum_{i=1}^N X_i}) \neq \sum_{i=1}^N(\frac{1}{ E X_i})$.

So how can I calculate this expected value then?

2. Re: expected value of division

I'm not sure if I understand your problem. Are the $X_i$ i.i.d. random variables whose distribution you know? Then I don't think there are many tricks to apply here. You need to express the expectation that you are trying to compute as an integral, and compute that integral. You won't be able to compute it simply from the expectation $E X_i$. One thing you can say though, is that by Jensen's inequality, if the $X_i$ are positive random variables,
$E \frac{1}{\sum_i X_i} \geq \frac{1}{\sum_i E X_i} = \frac{1}{N E X_i}$

3. Re: expected value of division

Yes, $X_i$ are iid (with $\Gamma (\alpha,\beta)$).

4. Re: expected value of division

Since your $X_i$ are Gamma-distributed, you are lucky, because

$\sum_i X_i \sim \Gamma(N\alpha,\beta)$

So basically, you only need to evaluate the integral

$\int_0^\infty \frac{1}{x} \gamma_{(N\alpha,\beta)}(x) dx$

where $\gamma_{(N\alpha,\beta)}(x)$ denotes the probability density function of a $\Gamma(N\alpha,\beta)$-distributed variable.

5. Re: expected value of division

Yes, thank, I have came to the same conclusion