# If two random variables have the same distribution, how to show after subtracting a common r.v., they still have the same distribution?

The question goes, if $$X, Y, Z$$ are independent random variables, where $$Z$$ is normal, and $$X+Z, Y+Z$$ have the same distribution, show $$X, Y$$ have the same distribution. I don't know whether and how generating functions can help, because $$X+Z$$ and $$Z$$ are not independent, and I cannot argue by subtracting $$Z$$ from $$X+Z$$. Could anyone please give any hint?

More generally, is it true when we don't assume $$Z$$ is normal?

• This is not true. in the general case. There is a counter-example in Feller's book. Nov 2, 2022 at 5:06

An easy proof when $$Z$$ is normal is using characteristic functions. Let $$f,g,h$$ be the characteristic functions of $$X,Y$$ and $$Z$$ respectively. Then $$f(t)h(t)=h(t)g(t)$$ and $$h(t)\neq 0$$ for all $$t$$ so $$f=g$$ which implies that $$X$$ and $$Y$$ have the same characteristic function.
• So normality of $Z$ can be replaced with "has a nonzero characteristic function"? Nov 2, 2022 at 15:22
If $$X$$ and $$Y$$ have the same distribution, can you show that $$X + Z$$ and $$Y + Z$$ have the same distribution for $$Z$$ normal with $$X,Y$$ and $$Z$$ independent? Call this Claim 1.
If so, since $$X + Z$$ and $$Y + Z$$ have the same distribution, add the normal distribution $$-Z$$ to $$X + Z$$ and $$Y + Z$$, by Claim 1 the resulting distributions are the same (namely, $$X$$ and $$Y$$ are the same).