If $X\sim\mathrm{exp}(1)$ and $Y\sim\mathrm{exp}(1),$ is $X=Y$?

Question

If two random variables have the same distribution it doesnt automatically mean that they are equal.

If $X\sim N(0,1)$ and $X=-Y,$ so is $Y\sim N(0,1).$ Because of the symmetry.

But if $X\sim\mathrm{exp}(1)$ and $Y\sim\mathrm{exp}(1).$ Could $X$ be not equal $Y$?

But is it possible, that the variables are almost surely different and have the same exp(1) distribution? — RCP9, Jun 23 '20 at 16:21

score 1 · Accepted Answer · answered Jun 23 '20 at 16:27

That two random variables have the same distribution doesn't guarantee they're equal. They could even be independent. All the matching distribution means is $x\le x,\,Y\le x$ have equal probabilities for all $x$, not that they have equal truth values.

score 1 · Answer 2 · answered Jun 23 '20 at 16:34

Two identically distributed random variables are (almost always) unequal. Otherwise the law of large numbers for IID (independent identically distributed) random variables would not be such a big deal.

When we say $$ \frac{X_1+X_2+\dots+X_n}{n} \to \mu $$ we do not mean $$ \frac{X_1+X_2+\dots+X_n}{n} = \frac{X_1+X_1+\dots+X_1}{n} = X_1 $$

Successive rolls of a die may be IID, but that does not mean that the die always comes up the same. Successive outcomes of the lottery may be IID, but what would be the point of a lottery if we know this week the winning number will be the same as last week?

If the common distribution has any probability masses, the variables will agree with positive probability, rather than being almost always unequal. — J.G., Jun 23 '20 at 17:48

score 0 · Answer 3 · answered Jun 23 '20 at 16:53

It is even possible that the two random variables are not defined on the same probability space so that they cannot be compared and statements like $X=Y$ make no sense at all.

Also if they are defined on the same space then they do not have to equal.

If $X\sim\mathrm{exp}(1)$ and $Y\sim\mathrm{exp}(1),$ is $X=Y$?

3 Answers3