# Combining two pdf

I would appreciate some insights on this question I've had this past days.

I'll try to phrase the problem as best as I can. I'll try to make some sense of it by considering a sales context.

Consider some sales time horizon $$t\in \left [ 0,\infty \right ]$$. The sales over time $$X$$ for some product given some condition $$y$$ ( day of the week for example ) follows a clearly defined pdf. Let's imagine that the sales over time $$X$$ for the same product given a condition $$z$$ ( weather ) also follows a clearly defined pdf.

If we have $$p_{X}(t|y)$$ and $$p_{X}(t|z)$$ can we build a third pdf i.e. $$p_{X}(t|y,z)$$ using this two pdf. This combined pdf would give us the probability of purchase for some day of the week and some specific weather condition ?

For simplicity we can assume that $$p_{X}(t|y)$$ and $$p_{X}(t|z)$$ belong to the same family i.e. they are both normal or binomial distributions or whatever they may be but they have an analytical form.

I've tried to explore a solution with the Bayes Theorem but I don't seem to get anywhere.

Just need to make sure $$\int{f_{X|Y,Z}(t|y,z)f_Z(z)dz}=f_{X|Y}(t|y)$$ and $$\int{f_{X|Y,Z}(t|y,z)f_Y(y)dy}=f_{X|Z}(t|z)$$.