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I am currently trying to solve a problem in the context of a Bayesian analysis that concerns normal distributions. The situation is as follows.

I have an equation that looks like this, where I know that ROE is normally distributed; both mean and variance are known, let's label them as $ROE_T$ and $\sigma$²:

$$LTS_0 = \frac{1}{(1+r_e)^{T-1}} \bigg(\frac{(ROE - r_e)B_{T - 1}}{r_e - ROE(1 - k_L)} \bigg)$$

All other variables are constant and do not depend on ROE.

From "merging" and rearranging the denominators of the equation and by assuming $T = 2$, I know that the distribution of the numerator and denominator will both also be normal with the following means and variances.

Numerator:

$$\begin{align} Numerator &\sim \mathrm{Normal}((ROE_T - r_e)B_{T - 1},\ \sigma^2 B_{T - 1}^2) \end{align}$$

Denominator:

$$\begin{align} Denominator &\sim \mathrm{Normal}(ROE_T w +r_e(1 + r_e),\ \sigma^2 w^2), \end{align}$$

where $ w = (k_L-1)(1+r_e)$, with $w < 0 $ because $ 0 < k_L < 1$ and $r_e > 0 $

I'm looking for the distribution of LTS given that ROE in both the numerator and denominator follows the abovementioned normal distributions. I know the basic math concerning random variables and expectational values, but I can't figure this out somehow, especially because there are normal distributions in the numerator and denominator and I cannot assume that they are independent (in fact, their correlation is perfectly negative since w is always negative and both are determined by ROE).

I have looked through articles including Hinkley (1969) and tried to apply the pdf displayed there. The Wikipedia page on ratio distributions has a guide on doing this, but everything I found is not applicable because of the perfect anticorrelation (often, specific denominators and variances become zero in that case). I've been trying to find a solution for a couple weeks now and I am starting to become a little bit desperate.

Simulation yields curves that are definetely not normally distributed.

I hope it actually exists, maybe someone knows exactly how I could derive it in this setting? I would be very thankful for any guidance!

Masel
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    You appear to overcomplicate the situation. Isn't $LTS_0$ just a (differentiable) function of $ROE$? That means you are asking how to transform a Normal random variable. That's a standard problem with a conventional solution (change of variables in the integration). Searching our site for "Jacobian" and "Normal" ought to produce many examples for you to learn from. Exactly how one carries out the calculation depends on the specific values of the Normal parameters and the constants involved in the transformation, which makes this question difficult to answer in full generality. – whuber Jan 29 '21 at 16:41
  • @whuber it is likely that I overcomplicate the situation. Maybe a little background information: I'd like to obtain closed-form distribution for LTS on a firm-specific level, where ROE determines the uncertainty with which LTS is estimated. I estimate the distribution of ROE using bayesian updating and the normal distribution of ROE then enters LTS to check the probability of firm value being smaller than the current price. Without a closed-form solution for the pdf of LTS, I will not be able to calculate the integrals i need for that. Is your idea easier than the one I had in mind? – Masel Jan 29 '21 at 17:09
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    If I understand your notation correctly ($ROE$ and $ROE_T$ are synonyms for a Normal random variable and everything else is a number), algebraic re-arrangement shows $LTS_0$ is obtained from the reciprocal of a Normally distributed variable (times a constant, plus another constant). Its density is given in closed form at https://stats.stackexchange.com/a/221252/919. – whuber Jan 29 '21 at 17:58
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    @whuber thank you for your helpful comments! – Masel Jan 30 '21 at 12:06

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