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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.

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There are different ways to construct the third pdf; it may not be unique.
Just need to make sure $\int{f_{X|Y,Z}(t|y,z)f_Z(z)dz}=f_{X|Y}(t|y)$ and $\int{f_{X|Y,Z}(t|y,z)f_Y(y)dy}=f_{X|Z}(t|z)$.

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  • $\begingroup$ Thank you for your response. I could try to solve both of these integral equations in order to find the combined pdf. $\endgroup$
    – abinos
    Commented Jan 25, 2021 at 14:51

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