Editing Normal distribution (section)

===== Operations on two independent normal variables =====
* If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two [[independence (probability theory)|independent]] normal random variables, with means <math display=inline>\mu_1</math>, <math display=inline>\mu_2</math> and variances <math display=inline>\sigma_1^2</math>, <math display=inline>\sigma_2^2</math>, then their sum <math display=inline>X_1 + X_2</math> will also be normally distributed,<sup>[[sum of normally distributed random variables|[proof]]]</sup> with mean <math display=inline>\mu_1 + \mu_2</math> and variance <math display=inline>\sigma_1^2 + \sigma_2^2</math>.
* In particular, if {{tmath|X}} and {{tmath|Y}} are independent normal deviates with zero mean and variance <math display=inline>\sigma^2</math>, then <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent and normally distributed, with zero mean and variance <math display=inline>2\sigma^2</math>. This is a special case of the [[polarization identity]].<ref>{{harvtxt |Bryc |1995 |p=27 }}</ref>
* If <math display=inline>X_1</math>, <math display=inline>X_2</math> are two independent normal deviates with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, and {{tmath|a}}, {{tmath|b}} are arbitrary real numbers, then the variable <math display=block>
    X_3 = \frac{aX_1 + bX_2 - (a+b)\mu}{\sqrt{a^2+b^2}} + \mu
</math> is also normally distributed with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>. It follows that the normal distribution is [[stable distribution|stable]] (with exponent <math display=inline>\alpha=2</math>).
* If <math display=inline>X_k \sim \mathcal N(m_k, \sigma_k^2)</math>, <math display=inline>k \in \{ 0, 1 \}</math> are normal distributions, then their normalized [[geometric mean]] <math display=inline>\frac{1}{\int_{\R^n} X_0^{\alpha}(x) X_1^{1 - \alpha}(x) \, \text{d}x} X_0^{\alpha} X_1^{1 - \alpha}</math> is a normal distribution <math display=inline>\mathcal N(m_{\alpha}, \sigma_{\alpha}^2)</math> with <math display=inline>m_{\alpha} = \frac{\alpha m_0 \sigma_1^2 + (1 - \alpha) m_1 \sigma_0^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math> and <math display=inline>\sigma_{\alpha}^2 = \frac{\sigma_0^2 \sigma_1^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math>.