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

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"?

Answer 1

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!

Answer 2

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 $\text{cor}(X,Y)=0$