The difference between "identically distributed" and having "common probability space" Suppose that we have two random variables X and Y. 
If we say they have a common probability space, does it mean that they are identically distributed? What is the difference between "identically distributed" and having "common probability space"?
Would you please give me an example of two random variables having the same probability space and but are not identically distributed, and another example of two random variables identically distributed and but they do not have the same probability space.
Thanks.
 A: Since you asked for two specific sorts of examples, here are some small ones.  First, consider a sample space consisting of two elements $a$ and $b$, with the probability measure that gives both points probability $\frac12$.  Consider the following two random variables $X$ and $Y$, both defined on this sample space. $X(a)=X(b)=Y(a)=0$ and $Y(b)=5$.  Then the probability distributions of $X$ and $Y$ are very different, so they are not identically distributed.
Second, consider a probability space consisting of four elements, $a,b,c,d$, each with probability $\frac14$. Let $Z$ be the random variable on this space defined by $Z(a)=Z(b)=0$ and $Z(c)=Z(d)=5$.  Then this $Z$ and the $Y$ from the previous paragraph are identically distributed, but they are defined on different sample spaces.
(If you want really small examples, both sorts of examples can be built using $1$-element sample spaces, but then there's not much probability or randomness visible.)
A: Nope. The same sample space says nothing about identically distributed.
I get the feeling you do not know what a sample space, if so, you need to look this up.
only when they live on the same sample space, it is makes sense to talk about dependence/independence but this has nothing to do with distributions being the same or different.
Example: Let $(\Omega_1,F,P)$ be a probability space with a random variable $X$ and $(\Omega_2,G,Q)$  be a probability space with random variable $Y$. $X$ and $Y$ can have the same distribution
Let $(\Omega_1\times\Omega_2,F\times G,P\times Q)$ be the product probability space of the previous two spaces, then $X$ and $Y$ have the same distribution and live on the same sample space.
but you could have made $X$ and $Y$ to have different distribution, the same would occur.
conclusion: same/different sample space says nothing about distribution
A: A probability space includes more information than a sample space. Whereas a sample space is all of the possible outcomes, it does not include the probability of each of the outcomes.
Consider a biased coin vs an unbiased coin. These have different associated probability spaces but the same sample spaces. They have the same possible outcomes (heads or tails) but different probabilities associated -- hence different distributions.
See: http://en.wikipedia.org/wiki/Probability_space
