find an example for random variables X,X',Y,Y' with the following characteristics:
a)X and X' have the same distribution
b)X+Y and X'+Y' have the same distribution
c)Y and Y' do not have the same ditribution
To find an example for random variables X, X', Y, and Y' with the specified characteristics, we need to define the distributions for each variable. Here's one possible example:
a) X and X' have the same distribution:
Let's assume X and X' are both independent random variables and have the same distribution. For simplicity, let's say they are both uniformly distributed between 0 and 1.
b) X+Y and X'+Y' have the same distribution:
To satisfy this condition, we need to choose the distribution of Y and Y' carefully. One way to achieve this is by letting Y be an independent random variable with a uniform distribution between -1 and 0. Similarly, let Y' be an independent random variable with a uniform distribution between -2 and -1. By choosing different ranges for the distributions of Y and Y', we ensure that the sums X+Y and X'+Y' have the same distribution.
c) Y and Y' do not have the same distribution:
As mentioned above, let Y be uniformly distributed between -1 and 0, and Y' be uniformly distributed between -2 and -1. Since their ranges are different, they do not have the same distribution.
In summary, using this example:
a) X and X' have the same distribution (uniformly distributed between 0 and 1).
b) X+Y and X'+Y' have the same distribution due to carefully choosing different ranges for Y and Y' while keeping X and X' unchanged.
c) Y and Y' do not have the same distribution due to their different ranges.