Adding Independent Normals Is Normal

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Much of math is a mystery to me. So is the following in that I have no idea how they came up with this attack.

Of course the math we learn in school took centuries to develop. But still, sometimes humans can be shockingly clever, especially compared to me.

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Claim

If $X_i$ for $i=1,2,...,n$ are independent and Normal RVs with parameters $\mu_i$ and $\sigma_i^2$ then the sum of all the $X_i$ is also Normal with parameters $\sum_{i=1}^{n} \mu_i$ and $\sum_{i=1}^{n} \sigma_i^2$

Proof

1. Begin by showing that if $X \sim N(0,\sigma_1^2)$ and $Y \sim N(0,1)$ then $X+Y \sim N(0,1+\sigma_1^2)$

This will payoff later down in step 6. Now you do this with the convolution formula. Get $f_{X+Y}(a)$ and see that that is the Normal pdf with parameters mentioned above, $0$ and $1+\sigma_1^2$

2. Now take a look at $X_i$ for $i=1,2$ which are INDEP Normal RVs with parameters $\mu_i$ and $\sigma_i^2$

3. Note that all the following shit cancels out so the LHS = RHS $$X_1+X_2 = \sigma_2(\frac{X_1-\mu_1}{\sigma_2} + \frac{X_2-\mu_2}{\sigma_2}) + \mu_1 + \mu_2$$

4. Now recall the transformation of a Normal RV. I already did this when showing how the Z table works. Recall that:

$$Y=aX+b$$ $$ \Rightarrow Y \sim N(a\mu + b, a^2 \sigma^2)$$

5. Therefore if we look at the RV Q which is the inside fraction part of the RHS of $X_1 + X_2$ that is,

$$Q = \frac{X_1-\mu_1}{\sigma_2} + \frac{X_2-\mu_2}{\sigma_2}$$ $$= (\frac{1}{\sigma_2} X_1 - \frac{\mu_1}{\sigma_2}) + (\frac{1}{\sigma_2} X_2 - \frac{\mu_2}{\sigma_2})$$

we notice that the first fraction of Q is $\sim N(\frac{1}{\sigma_2} \mu_1 - \frac{\mu_1}{\sigma_2}, \frac{1}{\sigma_2^2} \sigma_1^2) \sim N(0,\frac{\sigma_1^2}{\sigma_2^2})$

and the second fraction is $\sim N(0,1)$

6. From part 1 above, this means that

$$Q \sim N(0,1+\frac{\sigma_1^2}{\sigma_2^2})$$

7. And then note that

$$X_1 + X_2 = \sigma_2 Q + \mu_1 + \mu_2$$

Where $a = \sigma_2$ and $b = \mu_1 + \mu_2$ and mean is 0 and variance is $1+\frac{\sigma_1^2}{\sigma_2^2}$

Which implies that

$$X_1+X_2 \sim N(\sigma_2*0 + \mu_1 + \mu_2, \sigma_2^2 (1+ \frac{\sigma_1^2}{\sigma_2^2}))$$ $$\sim N(\mu_1 + \mu_2, \sigma_1^2+\sigma_2^2)$$

8. The general case for $n$ Indep and Normal RVs is proved through induction

9. Base case is $n=1$ which is true

10. Now assume it is true for $n-1$ and show for $n$

11. Note that

$$\sum_{i=1}^n X_i = \sum_{i=1}^{n-1} X_i + X_n $$

12. We just proved it for $n=2$ so we know that the whole thing is true or some ****.