sum of normal random variables

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Suppose \(X,Y\) are independent random variables from normal distributions, then sum of these two random variables is also a normal random variable. The proof of this statement could be referred to

Wikipedia-Proof using convolutions

We could also generate graphs to elucidate this process

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> x <- rnorm(10000,1,10)
> y <- rnorm(10000,-10,10)
> par(mfrow = c(3,1))
> plot(density(x))
> plot(density(y))
> plot(density(x+y))

hi

Reference