Question

If, X1, X2 be two independent normal variables with parameters (m1,s^2) and (m2,s'^2), obtain the distribution of (X1+X2). How do i do this?

Answer 1

Compute the convolution. The probability density function of those variables is
\[f_{X_1}(x) = \frac{1}{\sqrt{2\pi s^2}}e^{-\frac{(x-m_1)^2}{2s^2}}\]\[f_{X_2}(x) = \frac{1}{\sqrt{2\pi s'^2}}e^{-\frac{(x-m_2)^2}{2s'^2}}\]and the density of the sum of two variables is the convolution of those functions:\begin{eqnarray*}f_{X_1 + X_2}(x) &=& (f_{X_1} \ast f_{X_2})(x) \\
&=& \int_{-\infty}^{\infty}f_{X_1}(x - \xi)f_{X_2}(\xi)d\xi\\
&=& \frac{1}{2\pi ss'}\int_{-\infty}^{\infty}e^{-\frac{(x-\xi-m_1)^2}{2s^2}-\frac{(\xi-m_2)^2}{2s'^2}}d\xi\\
&=&\frac{1}{2\pi ss'}\frac{e^{-\frac{(x-(m_1+m_2))^2}{2(s^2+s'^2)}}\sqrt{2\pi}}{\sqrt{\frac{1}{s^2}+\frac{1}{s'^2}}}\\
&=&\frac{1}{\sqrt{2\pi(s^2+s'^2)}}e^{-\frac{(x-(m_1+m_2))^2}{2(s^2+s'^2)}} \end{eqnarray*}
which is a normal variable with parameters \[m_1+m_2\]and\[s^2+s'^2.\]

Answer 2

can i also do it by using the moment generating function?