What does it mean that $X$ and $Y$ have same probability distribution?

Question

If anyone says, $X$ and $Y$ have same probability distribution, then what does it mean?

I know if $X$ follows Binomial distribution, the $Y$ also follows Binomial distribution. But for being same probability distribution, do $X$ and $Y$ need to have same parameters too? That is, if $X\sim Binomial(20,0.6)$, then does $Y$ also require to be $Y\sim Binomial(20,0.6)$ for holding the condition that $X$ and $Y$ have same probability distribution ?

Also does "$X$ and $Y$ have same probability distribution" imply that "$X$ and $Y$ are independent"?

score 2 · Accepted Answer · answered Jun 28 '18 at 10:41

2

Taken as a general statement, I would say the same probability distribution means

$$Pr (X\leq t)=Pr (Y\leq t)$$

For this to be true the parameters need to be equal.

Regarding your last question, this is false, and can be shown by considering the special case $X=Y$. The above is true for this case, so same distribution, but clearly they are not independent!

answered Jun 28 '18 at 10:41

probabilityislogic

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+1 for ninja'ing me. I think it's funny that our answers are virtually identical and came in within two minutes of each other. – Josh Jun 28 '18 at 10:45
That is, if $X$ and $Y$ have same distribution, then they are *identically distributed*, but NOT necessarily *independent*. Have I stated it correctly? – user149054 Jun 28 '18 at 10:51
Yes that is correct – probabilityislogic Jun 28 '18 at 12:16

score 0 · Answer 2 · answered Jun 28 '18 at 10:43

1) Yes, I think it means that $X$ and $Y$ come from the same family and share the same parameters. I can't see how it would be useful to say that they have the "same" distribution unless this were the case.

2) Definitely not. Consider the case in which $X=Y$. It is also possible for two vectors $X$ and $Y$ to be drawn from the same distribution and be mutually orthogonal; i.e., $(X-\mu_x)^T(Y-\mu_y)=0$, or $\text{cor}(X,Y)=0$

What does it mean that $X$ and $Y$ have same probability distribution?

2 Answers2