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
ANSWERED
asked 2021-06-07

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

Answers (1)

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
\((X+Y)\sim N(\mu_{1}+\mu_{2}, \sigma_{1}^{2}+\sigma_{2}^{2})\)

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