How to add standard deviation regarding MATLAB function normrnd(mu,sigma)?

Question

My question depend on this scenario which is as follows,

I used a MATLAB function "normrnd(mu,sigma)" with mean 'mu = 0' and S.D 'sigma = 5', to generate a normal random number "R1". I added this random number "R1" to a number " X " which becomes X' = X + R1.

Now I again used normrnd to generate a normal random number, "R2", with mean = 0 and Sigma = 3. And again I add this number to X' which becomes X_new = X + R1 + R2.

So my question is if I use function normrnd(0,8) will it be same as using normrnd(0,5)+normrnd(0,3)?

Or in other word Can I add sigma using simple addition or is there any special rule to add standard deviations?

normrnd is a MATLAB function which generates random number from normal distribution. More info on normrnd is on the following link: http://www.mathworks.de/de/help/stats/normrnd.html

Thanks!

Answer 1

In general, a Gaussian random variable with mean $\mu$ and standard deviation $\sigma$ can be generated form a normal random variable $\zeta$ by $X = \mu+\sigma \zeta$.

So you have

$$R_1 = 0+5\zeta_1,\\ R_2 = 0+3\zeta_2.$$

Therefore, since $\zeta_1 \sim \zeta_2$, you have $R_1+R_2 \sim 8 \zeta$.