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I was trying to prove this statement in a formal way although intuitively it sounds true. is there any better and fast way to conclude this other than trying to see if the cumulative distribution function of $X, Z$ is equal to that of $X, Y$?

Firas Abd El Gani
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    See https://math.stackexchange.com/q/8742/321264, https://math.stackexchange.com/q/443659/321264. – StubbornAtom May 07 '20 at 20:51
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    The cdf of $(X,Z)$ is clearly not the same as the cdf of$(X,Y)$. It's easy to show $X$ amd $Z$ are independent, using the definition of independence. – David C. Ullrich May 07 '20 at 21:05

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