# Joint distribution from differential equations

I have the following problem -

Z is a random variable which can take any real value in the range [0,1]

a and b are independent variables drawn from uniform distribution in the interval [0,1].

Z is a function of a and b - Z(a, b)

I have differential equation for $$\frac{dZ(a, b)}{da}$$ and $$\frac{dZ(a, b)}{db}$$. I computed partial Z(a, b) (partial because they just reflect change in Z with respect to one variable). Let's denote them by $$Z_{a}(a, b)$$ and $$Z_{b}(a, b)$$

I have to find complete Z(a, b)

Will it be -

Z(a, b) = $$Z_{a}(a, b)$$*$$Z_{b}(a, b)$$ since a and b are independent

Or

Z(a,b) = prob(a)* $$Z_{a}(a, b)$$ + prob(b)* $$Z_{b}(a, b)$$

I am confused. Any clarification will be helpful. Thank you.

• Please clarify: it is not possible to write in turn $Z(a), Z(b), Z(a,b)$. Is there any difference between these functions? If yes, please give their definitions and use different notations for different functions. Aug 30, 2022 at 14:17
• @Bertrand thanks for the reply. Basically, I denoted Z(a) because I computed it from differential equation $\frac{dZ(a, b)}{da}$. Z is still a function of both a and b. Is that incorrect way to denote? Aug 30, 2022 at 16:34
• So, let me explain my problem. I recovered Z(a, b) from two differential equation. But overall Z(a, b) will be an expectation computed from combining the two. How should the combination be is my question. Hope that clarifies Aug 30, 2022 at 16:40
• cross posted at math.stackexchange.com/questions/4521484/… Aug 30, 2022 at 23:36

From the fundamental theorem of calculus it follows that the relationship between a function and its partial derivative is given by: $$Z(a,b)= \int_0^a \frac{\partial Z}{\partial a}(x,b) dx + Z(0,b),$$ and similarly for the partial derivative of $$Z$$ wrt $$b$$.