# Conditional expected value

Hi, I'm not very familiar with statistics but maybe someone here can help me:

Let $X$ denote a random variable. At the beginning of period 1, I assume that $X \sim \mathcal{N}(m_0, \sigma^2)$

After the first period I observe $a$ with $a = \frac{b + cX}{1 + X} + Y$ with $Y \sim \mathcal{N}(0, \sigma_Y^2)$ and $b, c$ being known parameters.

I now want to update my beliefs about the distribution of $X$

Rearranging: $X = \frac{a - b - Y}{-a + c + Y}$

So basically I'm looking for the expected value $E[\frac{a - b - Y}{-a + c + Y}|a, b, c]$ and the variance of $X$. Or (more general) $E[\frac{f - Y}{g + Y}|f, g]$

Is this possible? Is there a formula I can use?

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might help to know that $a/(b+Y)$ has generally no expectation (not integrable) –  Alekk Jun 21 '10 at 11:03
Even if the conditional expectation sought here does not exist, the condition distribution may exist (I'm going to see if I can parse the question before changing "may" to "does".) –  Michael Hardy Jun 21 '10 at 22:08