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I'm know through my text book ,that these property is valid for statistically independent variable:

property 1 for waited values

$$\mathsf E\left[\prod\limits_{i=1}^n g_i(x_i)\right]=\prod\limits_{i=1}^n \mathsf E\left[g_i(x_i)\right]$$ property 2 for probability density function(p.d.f)

$$p_{x,y}(X,Y)=p_x(X)\,p_y(Y)$$

but in one question, two p.d.f from x and y random variables was given to me : figure 3 $$\begin{align}p_x(X)&=ae^{-aX}u(X)\\[1ex]p_y(Y)&=be^{-bY}u(Y)\end{align}$$

If I had the $p_{x,y}(X,Y)$ function, I'd know previously if $x$ and $y$ are statistically independents variables, however I only have $p_x(X)$ and $p_y(Y)$, and I don't know how to conclude hat $x$ and $y$ are statistically independents. I one resolution of my university list, it's assumed, in case of figure 3, that x and y are statistically independents,but I don't know how and why. Could someone give me an idea??

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Indeed, no.

By themselves, those two marginal probability density functions are not enough to lay claim to independence.

There would need to be some other reason provided in the question for you to be able to justify that assumption.

However, without seeing the whole question, we cannot point to what that reason might be.

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  • $\begingroup$ the question gave this two random variables. X is about a equipment lifetime and Y is the equipment recovery. And there's a third variable Z= X + Y that model the cycle time. The question only give the meaning of each variable, but don't say nothing beyond that. $\endgroup$ May 14, 2020 at 17:23
  • $\begingroup$ note: the question didn't give pxy(X,Y). it's in portuguese and I can't translate the question very well. $\endgroup$ May 14, 2020 at 17:25
  • $\begingroup$ So is it reasonable, or not, to assume that equipment life time and equipment recovery time might be independent processes? $\endgroup$ May 14, 2020 at 23:02
  • $\begingroup$ yes, but it's an assumption. I'd be more sure if I could prove that x and y are statistically independent in a mathematically way. $\endgroup$ May 14, 2020 at 23:12

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