I have two variables 2X and 0.5Y, both are independent and follows Gaussian distribution. How to estimate their mean and variance analytically? I want to know their individual mean and variance, then I want to add them.
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Let $z = ax$ and $w=by$ with $a$ and $b$ known constants and $x\sim \mathcal N(\mu_x,\sigma_x^2)$ and $y\sim\mathcal N(\mu_y,\sigma_y^2)$ then
$$v = z + w = ax + by$$ is sum of two independent normally distributed variables and hence will be normal.
The mean $\mu_v = \mathbb E[v] = a\mu_x + b \mu_y$ and
The variance $\sigma^2_v = a^2 \sigma_x^2 + b^2\sigma_y^2$ since covariance part is 0 from the assumption of independence.