# Formalism to cope with probability density functions defined piecewise (in the second dimension)

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.

• You might wish to take a look at this: en.wikipedia.org/wiki/Mixture_distribution – Grada Gukovic Jul 17 at 15:33
• Work with the CDF instead or else frame everything in terms of conditional and marginal probabilities. – whuber Jul 17 at 17:24
• Thanks, that's indeed the term I needed to search for further references – Onturenio Jul 17 at 17:47
• I have solved my problem following your hint. You may wish to turn your comment into a question so we close the question as answered. – Onturenio Jul 20 at 11:54