Are combinations of independent random variables also independent

Question

I have three i.i.d. random variables: X, Y and Z. I have two constants: c1 and c2.

Is it true that: Pr[Y-X < c1 AND Z-X < c2] = Pr[Y-X < c1] * Pr[Z-X < c2]?

I am guessing that this only holds when Y-X and Z-X are independent.

Answer 1

By definition if $Y-X$ and $Z-X$ are independent, $$ P( (Y-X < c_1) \wedge (Z-X < c_2) ) = P(Y-X < c_1) \cdot P(Z-X < c_2) $$

But with $\{X, Y, Z\}$ i.i.d. random variables, it is hard to find an underlying distribution such that $Y-X$ and $Z-X$ are independent. In fact, the only case where this would be true would be if the underlying distribution has all its probability at one value!

To show that $Y-X$ and $Z-X$ need not be independent, consider $\{X, Y, Z\}$ to be independent uniform randoms on $(0,1)$.

Then take $c_1 = +0.9$ and $c_2 = -0.2$. Now whenever $Y-X < c_1$ is false, $Z-X < c_2$ must also be false. Since neither of those statements are always false, in this case $Y-X$ and $Z-X$ are not independent.