# What is the distribution of the sums of two independent distributions?

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 $$X+Y$$?

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

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.