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Problem:

Random variables X, Y are independent. X has normal distribution with parameters (a= 1, b^2 = 4). Y has also normal distribution, but Y is equal to (-X). Find all constants c and k > 0,in order to find X + Y + c and k*X so they would have same distribution.

I found out, it has to be done throught convolution theorem. So first I will sum two random variables and then compute constant c. But I don't really know how to compute the constant. My idea is to use definite integral of f(z) + c = 1 (f(z) is X+Y). Is it correct?

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  • $\begingroup$ Sorry, but how can $X,Y$ be independent if $Y=-X$? Do you mean that it has the same distribution as $-X$? $\endgroup$ – Clement C. Dec 11 '16 at 19:53
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Please see solution in the image. You don't need to perform convolution as The sum of Gaussian RV's is also Gaussian.

solution

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  • $\begingroup$ Don't sweat it. Notice that I equated only the first two moments. Is it enough to say that the whole distribution is identical? Think about it.... $\endgroup$ – Nir Regev Dec 11 '16 at 19:54
  • $\begingroup$ Okay, I will think about it. $\endgroup$ – Mafi Dec 11 '16 at 19:57
  • $\begingroup$ Also: important: X,Y are independent. So X does not equal -Y. You probably meant that they have opposite means. Check the question again. $\endgroup$ – Nir Regev Dec 11 '16 at 20:01
  • $\begingroup$ Please, why these two moments equals? $\endgroup$ – Mafi Dec 11 '16 at 21:44
  • $\begingroup$ Gaussian distribution is completely defined by its first two moments. $\endgroup$ – Nir Regev Dec 11 '16 at 21:48

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