Normal distributions sums

Question

I read this property about normal distribution

If $X\sim\mathcal N(\mu_X,\sigma_X^2)$ and $Y\sim\mathcal N(\mu_Y,\sigma_Y^2)$ are independent, then $$ X+Y\sim\mathcal N(\mu_X+\mu_Y,\sigma_X^2+\sigma_Y^2). $$

However I also read $$ X = \sigma Z + \mu\, $$

If I sum X + Y using this property I get

$$ X+Y = (\sigma_x + \sigma_y) Z + \mu_x + \mu_y = N(\mu_x + \mu_y, (\sigma_x + \sigma_y)^2), $$

Why do I get different result on variance?

Answer 1

You are taking $X=\sigma_xZ+\mu_x$ and $Y=\sigma_yZ+\mu_y$. They are not independent.

You should take $X=\sigma_xZ_1+\mu_x$ and $Y=\sigma_yZ_2+\mu_y$ where $Z_1$ and $Z_2$ are iid with standard normal distribution.