X N( 1, 12) and Y N( 2, 22) are independent. What is the distribution of X+Y?

Linearity of the mean. E[X+Y]= 1+ 2 regardless of dependence. Variance under independence. Var(X+Y)= 12+ 22 since Cov(X,Y)=0.

Sum of Independent Normals

Statistics

Normal Distribution

Easy

X \sim N (μ_{1}, σ_{1}^{2})

and

Y \sim N (μ_{2}, σ_{2}^{2})

are independent. What is the distribution of

X + Y

Hint 1

Sums of independent normal random variables are again normal.

Hint 2

Means always add; for independent variables, variances also add.

Show worked solution

1. Linearity of the mean
$E [X + Y] = μ_{1} + μ_{2}$ regardless of dependence.
2. Variance under independence
$Var (X + Y) = σ_{1}^{2} + σ_{2}^{2}$ since $Cov (X, Y) = 0$ .
3. Conclude normality
A sum of independent normal variables is normal, so $X + Y \sim N (μ_{1} + μ_{2}, σ_{1}^{2} + σ_{2}^{2})$ .

Answer: N(mu1 + mu2, sigma1^2 + sigma2^2)