If&nbsp;$X\sim N\left( {{\mu }_{x}},{{\sigma }_{x}}^{2} \right)$ and $Y\sim N\left( {{\mu }_{y}},{{\sigma }_{y}}^{2} \right)$ are independent and $a$, $b$ are constants, then
•	$X\pm Y\sim N\left( {{\mu }_{x}}\pm {{\mu }_{y}},{{\sigma }_{x}}^{2}+{{\sigma }_{y}}^{2} \right)$
•	$aX+bY\sim N\left( a{{\mu }_{x}}+b{{\mu }_{y}},{{a}^{2}}{{\sigma }_{x}}^{2}+{{b}^{2}}{{\sigma }_{y}}^{2} \right)$
For&nbsp;$n$ independent trials of $X$,
•	 ${{X}_{1}}+{{X}_{2}}+...{{X}_{n}}\sim N\left( n{{\mu }_{x}},n{{\sigma }_{x}}^{2} \right)$
•	mean of $X$, $\bar{X}=\frac{{{X}_{1}}+{{X}_{2}}+...{{X}_{n}}}{n}\sim N\left( {{\mu }_{x}},\frac{{{\sigma }_{x}}^{2}}{n} \right)$

If $X\sim N\left( {{\mu }_{x}},{{\sigma }_{x}}^{2} \right)$ and  $Y\sim N\left( {{\mu }_{y}},{{\sigma }_{y}}^{2} \right)$ are independent and $a$, $b$ are constants, then

•	$X\pm Y\sim N\left( {{\mu }_{x}}\pm {{\mu }_{y}},{{\sigma }_{x}}^{2}+{{\sigma }_{y}}^{2} \right)$

•	$aX+bY\sim N\left( a{{\mu }_{x}}+b{{\mu }_{y}},{{a}^{2}}{{\sigma }_{x}}^{2}+{{b}^{2}}{{\sigma }_{y}}^{2} \right)$

•	 ${{X}_{1}}+{{X}_{2}}+...{{X}_{n}}\sim N\left( n{{\mu }_{x}},n{{\sigma }_{x}}^{2} \right)$

•	mean of $X$, $\bar{X}=\frac{{{X}_{1}}+{{X}_{2}}+...{{X}_{n}}}{n}\sim N\left( {{\mu }_{x}},\frac{{{\sigma }_{x}}^{2}}{n} \right)$

Properties of the Normal Distribution and Linear Combinations

Properties of the Normal Distribution and Linear Combinations

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