$y_1 = x_1 + x_0$;

$y_2 = x_2 + x_0$.

Suppose that $x_1$, $x_2$, and $x_0$ are independent with each other. They all follow the uniform distribution in $[0, 1]$.

Then, I want to know if $y_1$ and $y_2$ are independent and how to obtain the joint pdf? Thanks

  • $\begingroup$ Can you show some work? Can you think of a case when information about $Y_1$ can tell you something about $Y_2$? $\endgroup$ – Michael Oct 18 '18 at 4:27
  • $\begingroup$ Thanks for reply. I think there is a common term in y1 and y2. I'm not sure if y1 and y2 are independent. Thay may have some relation due to the common term. For example, if x1=0 and x2=0. Then y1 and y2 become dependent, although it is not the case that I asked. My concern is that because a common tem is involved in two random vaeiables, is there is theory to determine the dependence of y1 and y2?Thanks. $\endgroup$ – Quentin Oct 18 '18 at 5:16
  • $\begingroup$ You can easily compute $Ey_1, Ey_2$ and $Ey_1y_2$. You will see that $Ey_1y_2 \neq Ey_1Ey_2$, hence $y_1,y_1$ are not independent $\endgroup$ – Kavi Rama Murthy Oct 18 '18 at 6:44
  • $\begingroup$ Suppose I tell you that $Y_1=2$. Does that tell you anything about $Y_2$? $\endgroup$ – Michael Oct 18 '18 at 14:56
  • $\begingroup$ Thank you very much. @KaviRamaMurthy Rama Murthy How about if we further perform an operation such that if y1 is greater than 1 (similarly to y2), then we minus 1 to keep it still in the interval of [0 1]. This means that we assume y1 has a period of 1. Then, I do the simulations and find that the two variables y1 and y2 are uncorrelated, but are they independent? Thanks. $\endgroup$ – Quentin Oct 19 '18 at 3:55

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