PDF and CDF of sum of random variables with different distributions This may sound too trivial but I am having difficulty to solve an assignment problem where I need to determine the distribution and density of a random variable $Z$ which is the sum of random variables with different distributions.
In general, what is the technique to find the distribution and the density when there are random variables of different distribution?
 A: The density of the sum of two independent Beta distributions with possibly different parameters has been derived, but I am not sure you really want to see how it looks like. Adding a normal r.v. to the mix...
Meanwhile, one could ponder the normal approximation to the beta distribution (see @whuber's answer in this post, and also this blogpost ).
...in which case $Z$ will be approximately normally distributed, with
$$E(Z) = \frac {(1+\theta)\alpha}{\alpha+\beta},\;\; \text{Var}(Z) = \frac {(1+\theta^2)\alpha \beta}{(\alpha+\beta)^2(\alpha+\beta+1)} + \sigma^2_w$$
(assuming that the two betas have identical parameters).
But I would be thrilled if I am neglecting some known result or a clever trick, in which case I guess somebody will jump in and provide a neat PDF and a nice-looking CDF.
ADDENDUM
To obtain the density etc of the sum of independent random variables, a number of techniques are available. One is convolution. This involves integration, and care must be exercised when the support of the variables involved has bounded support.
Another method is through the use of the characteristic function/moment generating function, where a specific relation holds between the CF/MGF's of the variables. The hope is that the result will be an expression that is recognizable as the MGF of a known distribution.
