I'm not sure how to pose this question, as I lack the correct terminology. Actually, my question tries to obtain insight on the terminology and notation to cope with the following problem:
I have a probability density function that depends on two variables, one of which is discrete. In particular, the problem reads: if a given day is windy, the precipitation follows a given PDF $f_1(x)$. If it's not windy, then precipitations is distributed as $f_2(x)$. Now, it's known that the probability of a day to be windy is 0.7. The first problem is how to obtain the PDF of a generic day. I believe I can do nothing but to define the function piecewise such as:
$ f(x,v) = \left\{ \begin{array}{lr} 0.7\ f_1(x) & v=1\\ 0.3\ f_2(x) & v=0 \end{array} \right. $
But it's a rather convoluted notation hard to work with. For instance, in the next point I'm asked to calculate the mean and the variance of the precipitation. How could I write the PDF in a more convenient fashion? This must be a rather common problem, but I cannot find examples of PDFs defined in this way. I have been trying to google for similar examples, but as I lack the right terminology, I have found nothing so far.
Therefore, any suggestion on how to address this problem, or reference where this type of problems are formally discussed, is welcome.
