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.