operations on probability distributions I've found that you can do certain arithmetic operations on random variables such as :


*

*multiply or divide two log-normal distributed variables

*add or divide two gamma distributed variables


I've been searching on Internet and I found nearly nothing. I would like to compute the mean and variance for each of these operations. Do you know any website which give the formulas or do you have the answer ?
 A: This is a (partial) answer to the first item:
Let $X$ and $Y$ be two log-normally distributed random variables, and let
$$
\mathrm{E}(X) = \mathrm{e}^{\mu_X + \frac{1}{2}\sigma_X^2} \quad\text{and}\quad \mathrm{E}(Y) = \mathrm{e}^{\mu_Y + \frac{1}{2}\sigma_Y^2}
$$
be their means and
$$
\mathrm{Var}(X) = \mathrm{E}^2(X) \big( \mathrm{e}^{\sigma_X^2} - 1 \big) \quad\text{and}\quad \mathrm{Var}(Y) = \mathrm{E}^2(Y) \big( \mathrm{e}^{\sigma_Y^2} - 1 \big)
$$
their variances.
On the assumption that $X$ and $Y$ are independent, the product $X \cdot Y$ and the ratio $\frac{X}{Y}$ are also log-normally distributed. Their means are given by
$$
\mathrm{E}(X \cdot Y) = \mathrm{e}^{\mu_X + \mu_Y + \frac{1}{2}\left( \sigma_X^2 + \sigma_Y^2 \right)} \quad\text{and}\quad \mathrm{E}\left( \frac{X}{Y} \right) = \mathrm{e}^{\mu_X - \mu_Y + \frac{1}{2}\left( \sigma_X^2 + \sigma_Y^2 \right)},
$$
and their variances are
$$
\mathrm{Var}(X \cdot Y) = \mathrm{E}^2(X \cdot Y) \big( \mathrm{e}^{\sigma_X^2 + \sigma_Y^2} - 1 \big) \quad\text{and}\quad \mathrm{Var}\left( \frac{X}{Y} \right) = \mathrm{E}^2\left( \frac{X}{Y} \right) \big( \mathrm{e}^{\sigma_X^2 + \sigma_Y^2} - 1 \big).
$$
A: For your second question: the gamma distribution is closed under convolution. What this means is that the sum of (independent) gamma random variables is again gamma distributed (if they have the same scale parameter). You can easily show this with for example the moment generating function: 
Let $S = X + Y$ with $X \sim G(\alpha_X,\beta_X)$ and $Y\sim G(\alpha_Y,\beta_Y)$ independent of each other. Let $\beta_X=\beta_Y = \beta$. Then
\begin{equation*}
M_S(t) = E\bigl[e^{t(X+Y)}\bigr] = E\bigl[ e^{tX}\bigr]E\bigl[e^{tY}\bigr] = \left(\frac{\beta}{\beta-t}\right)^{\alpha_X}\left(\frac{\beta}{\beta-t}\right)^{\alpha_Y} = \left(\frac{\beta}{\beta-t}\right)^{\alpha_X+\alpha_Y},
\end{equation*}
so $S\sim G(\alpha_X+\alpha_Y,\beta)$. For the moment generating function of the gamma distribution, see e.g. the Wikipedia article on the gamma distribution.

If you take the fraction between two (independent) gamma r.v.'s you will get the product of a gamma and an inverse gamma, since
\begin{equation*}
Y \sim G(\alpha_Y,\beta_Y) \quad \Rightarrow \quad \frac 1Y \sim IG\left(\alpha_Y,\frac 1{\beta_Y}\right).
\end{equation*}
You can show this with the change of variable $g(y) = 1/y$. Then moments of $X/Y$ are not complicated because of the independence. For example the first moment,
\begin{equation*}
E\left[\frac XY\right] = E[X] \cdot  E\left[\frac 1Y\right] = \frac {\alpha_X}{\beta_X} \cdot \frac{1}{\beta_Y(\alpha_Y-1)}.
\end{equation*}
The distribution of $X/Y$, however, is another talk. I do not know what that would be and do not have any references for it.
