Question

# Compute the distribution of X+Y in the following cases:X and Y are independent normal random variables with respective parameters (\mu_{1},\sigma_{1}^{2}) and (\mu_{2},\sigma_{2}^{2}).

Random variables

Compute the distribution of $$X+Y$$ in the following cases:
X and Y are independent normal random variables with respective parameters $$(\mu_{1},\sigma_{1}^{2}) and (\mu_{2},\sigma_{2}^{2})$$.

2021-06-08

$$X\sim N(\mu_{1},\sigma_{1}^{2})$$
$$Y\sim N(\mu_{1}, \sigma_{2}^{2})$$
X and Y are independent.
$$N(\mu_{1},\sigma_{1}^{2})+N(\mu_{1},\sigma_{2}^{2})\sim N(\mu_{1}+\mu_{2},\sigma_{1}^{2}+\sigma_{2}^{2})$$
So,
$$X+Y$$ follows normal distribution
$$(X+Y)\sim N(\mu_{1}+\mu_{2}, \sigma_{1}^{2}+\sigma_{2}^{2})$$