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Part of my preparation for the exam of probability, I'm trying to create a sheet of useful theorems. I'm trying to figure out if there is a general theorem that state the distribution of $X+Y$ where $X$ have distribution one and and $Y$ have distribution another (they could be the same distribution) and they are independent.

For example, I know that if $X\sim Pois(\lambda_1)$ and $Y\sim Pois(\lambda_2)$ and they are independent then $(X+Y)\sim Pois(\lambda_1+\lambda_2)$. I'm interested in binomial, geometric, poisson, uniform, beta and gamma distributions. For example, what is the distribution of the sum of $X\sim Uni[a,b]$ and $Y\sim Uni[c,d]$. I guess it does not make sense to look at the sum of discrete and continuous random variables so it simplifies the question. What can we say about $X+Y$?

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  • $\begingroup$ If $X$ and $Y$ are not independent you cannot find the distribution of $X+Y$. If they are independent then $F_{X+Y}(z)=\int F_X(z-y)dF_Y(y)$. $\endgroup$ Aug 22, 2020 at 12:29
  • $\begingroup$ @KaviRamaMurthy Oh cool, what is the general version of the formula you wrote? Also how do you translate it to pdf? $\endgroup$
    – vesii
    Aug 22, 2020 at 12:30

2 Answers 2

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In the general case of two possibly dependent random variables, you need to know the joint density of $(X,Y)$ in order to get that of $X+Y$. You may calculate it using the transformation formula, looking at the map $g(x,y)=(x+y,y)$ and then integrating to get the marginal density.

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You can use the convolution of the distribution functions , in this case of X and Y to find the distribution function of the sum X+Y. The convolution of probability distributions corresponds to the distribution function of the addition of independent random variables and, by extension, to forming linear combinations of random variables. Furthermore, we have that the characteristic function of the sum of two independent random variables is the product of their characteristic functions. With these ideas, you can find the distribution function of X+Y and finally analize all their properties.

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