<p>From the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow noreferrer">fundamental theorem of calculus</a> it follows that the relationship between a function and its partial derivative is given by:
<span class="math-container">$$ Z(a,b)= \int_0^a \frac{\partial Z}{\partial a}(x,b) dx + Z(0,b), $$</span>
and similarly for the partial derivative of <span class="math-container">$Z$</span> wrt <span class="math-container">$b$</span>.</p>

CC BY-SA 4.0
2022-09-01T09:26:41.290Z
I couldn't understand the answer. What are you trying to do here and what is Z(0,b). Isn't this just getting Z(a, b) from individual differential equation. I have already done that. My question is how to combine the two Z(a, b) that I got.
Elina Gilbert
CC BY-SA 4.0
2022-09-01T08:37:48.697Z
2022-09-01T15:27:09.750Z
2022-09-01T15:27:09.750Z
Bertrand
<p>I have the following problem -</p>
<p>Z is a random variable which can take any real value in the range [0,1]</p>
<p>a and b are independent variables drawn from uniform distribution in the interval [0,1].</p>
<p>Z is a function of a and b - Z(a, b)</p>
<p>I have differential equation for <span class="math-container">$\frac{dZ(a, b)}{da}$</span> and <span class="math-container">$\frac{dZ(a, b)}{db}$</span>. I computed partial Z(a, b) (partial because they just reflect change in Z with respect to one variable). Let's denote them by <span class="math-container">$Z_{a}(a, b) $</span> and <span class="math-container">$Z_{b}(a, b) $</span></p>
<p>I have to find complete Z(a, b)</p>
<p>Will it be -</p>
<p>Z(a, b) =  <span class="math-container">$Z_{a}(a, b) $</span>*<span class="math-container">$Z_{b}(a, b) $</span> since a and b are independent</p>
<p>Or</p>
<p>Z(a,b) = prob(a)* <span class="math-container">$Z_{a}(a, b) $</span> + prob(b)*  <span class="math-container">$Z_{b}(a, b) $</span></p>
<p>I am confused. Any clarification will be helpful. Thank you.</p>

CC BY-SA 4.0
2022-08-30T14:17:43.507Z
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.
Bertrand
CC BY-SA 4.0
2022-08-30T16:34:37.960Z
@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?
Elina Gilbert
CC BY-SA 4.0
2022-08-30T16:40:23.357Z
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
Elina Gilbert
CC BY-SA 4.0
2022-08-30T23:36:39.550Z
cross posted at https://math.stackexchange.com/questions/4521484/finding-joint-distribution-from-marginal-distribution
Henry
<probability><difference-equations><expectations>
Joint distribution from differential equations
Elina Gilbert
# Joint distribution from differential equations

(Asked by: username_0 on 2022-08-30)

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.

**Tags:** probability, difference-equations, expectations
> **Comments:**
> **username_1**: 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.  
> **username_0**: @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?  
> **username_0**: 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  
> **username_2**: cross posted at https://math.stackexchange.com/questions/4521484/finding-joint-distribution-from-marginal-distribution  

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## 1 answers

### Answer by username_1 on 2022-09-01. Score: 1

From the [fundamental theorem of calculus](https://en.wikipedia.org/wiki/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 $b$.

> **Comments:**
> **username_0**: I couldn't understand the answer. What are you trying to do here and what is Z(0,b). Isn't this just getting Z(a, b) from individual differential equation. I have already done that. My question is how to combine the two Z(a, b) that I got.  

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