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January 8th, 2018, 04:00 AM  #1 
Newbie Joined: Jan 2018 From: Tunis, Tunisia Posts: 10 Thanks: 4 Math Focus: Analysis  Two random variables equal in distribution
Hi ! Can we find two random variables that are equal in distribution but are not equal ? Thanks in advance ! 
January 8th, 2018, 04:04 AM  #2 
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92 
Of course, we can even find such variables that are independent. The simplest example, take two similar coins. Throw them and see if you get head or tail. The distribution of the first coin will be the same as the distribution of the second. But they're surely not the same outcome. 
January 8th, 2018, 04:56 AM  #3  
Newbie Joined: Jan 2018 From: Tunis, Tunisia Posts: 10 Thanks: 4 Math Focus: Analysis 
Hi ! Quote:
 
January 8th, 2018, 10:40 AM  #4 
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92 
Let's take as sample space $\Omega = \{(0,0), (0,1), (1,0), (1,1)\}$, where $0$ is tail and $1$ is head. Then we can take as variable $X(p,q) = p$ and $Y(p,q) = q$. 
January 8th, 2018, 02:16 PM  #5 
Newbie Joined: Jan 2018 From: Tunis, Tunisia Posts: 10 Thanks: 4 Math Focus: Analysis  
January 8th, 2018, 04:42 PM  #6 
Senior Member Joined: Sep 2015 From: USA Posts: 1,790 Thanks: 922 
as a somewhat more extreme example suppose that the temperature in Miami is uniformly distributed between 60 and 100 degrees. (it's not but suppose it is). and let's further suppose that the age of retirees living in Miami is also uniformly distributed between 60 and 100 yrs old. Clearly these two random variables, temperature and age, are not equal, but they have the same distribution. 

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