Question

What is the distribution of X+Y if X ~ normal(500 m, 90 m) and Y ~ normal(5 km, 0.1 km) and X and Y are independent?

Answer 1

I think one way might be to examine the moment generating function of \(X+Y\):
\[\begin{align*}m_{X+Y}(t)&=E\bigg[e^{(X+Y)t}\bigg]\\
&=E\bigg[e^{Xt}\cdot e^{Yt}\bigg]\\
&=E\bigg[e^{Xt}\bigg]E\bigg[e^{Yt}\bigg]&&\text{since }X\text{ and }Y\text{ are independent}\\
&=m_X(t)\cdot m_Y(t)
\end{align*}\]
The mgf of a normally distributed random variable \(Z\) with mean \(\mu\) and standard deviation \(\sigma\) is \(m_Z(t)=\exp\left(\mu t+\dfrac{1}{2}\sigma^2t^2\right)\).