sum of two normally distributed random variables enter image description hereI know the theorem that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances. But when I look through the proof of this theorem on wikipedia, i find there's something hard for me to understand. Based on what I have learned so far, I can only use the convolution formula to complete this proof. Everything goes smoothly except for the last step. Suppose Z is X+Y.
I can't figure out why the integral on the right side is 1. I don't understand how each part of this integral corresponds to the basic form of the normal distribution.
https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
 A: I think you are referring to this proof:
https://en.m.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
This is a common trick in probability. To convince yourself, take the object they claim integrates to 1 and “spell it out” as a normal density function. To do this, remember a normal density function will have some variance $\tilde{\sigma}$ and mean $\tilde{\mu}$. Define these variance and mean in order for the expression to look “simpler”, the variance $\tilde{\sigma}$ will need to be a ratio of the variance of $X$ times the variance of $Y$, and the variance of $Z$.
Define a new normal random variable $W$,(because, why can’t you do that? I can take any real number $>0$ and call that the variance of a new random variable and any real number, even zero or negative, and call that the mean) with variance $\tilde{\sigma}$ and mean $\tilde{\mu}$ and write down the property that if you integrate over the whole of the real numbers the density of $W$ gives you 1.
Give it a shot and if you need more help send a comment.
