# If two random variables are defined on the same probability space do they have the same distribution?

Let's say we have two random variables $$X$$ and $$Y$$ both defined on the probability space $$(\Omega,\mathcal{F},\mathbb{P})$$. So $$X$$ and $$Y$$ are measurable functions from $$\Omega$$ to $$\mathbb{R}$$.

Do $$X$$ and $$Y$$ necessarily have the same distribution? Can we for example have $$X\sim Bern(p)$$ and $$Y\sim N(0,1)$$? Or would we then say that $$X$$ and $$Y$$ have for example the measure $$\mathbb{P}_X$$ and $$\mathbb{P}_Y$$ respectively?

You can easily construct such $$X$$ and $$Y$$. Take $$(\Omega,\mathcal{F},\mathsf{P})=((0,1),\mathcal{B}_{(0,1)},\lambda)$$, where $$\lambda$$ is the Lebesgue measure, and set $$X(\omega):=1\{\omega\le p\},$$ and $$Y(\omega):=\Phi^{-1}(\omega),$$ where $$\Phi$$ is the standard normal cdf. Then $$\mathsf{P}(X=1)=\lambda(\{\omega\le p\})=p,\quad \mathsf{P}(X=0)=1-p,$$ and $$\mathsf{P}(Y\le y)=\lambda(\{\Phi^{-1}(\omega)\le y\})=\lambda(\{\omega\le \Phi(y)\})=\Phi(y).$$ That is, $$X\sim\text{Bern}(p)$$ and $$Y\sim N(0,1)$$.
An interesting related question is whether one can construct independent random variables on the same probability space (s.t. $$P_{X,Y}=P_X\otimes P_Y$$).
No, they do not. For example, the random variable on the interval with the uniform measure which maps everything to $$1$$ has a delta function distiribution, while the one that maps $$x$$ to $$x$$ has uniform distribution, while the one that maps $$x$$ to $$x^2$$ has some non-uniform distribution.
Simple example: Two coins, one biased $$P(heads)=.75$$ and the other not $$P(heads)=.5$$. Same sample spaces for both.