Given two independent normal distributions $X\sim N(10,12)$ and $Y\sim N(5,8)$. Find the normal

Question

Given two independent normal distributions $X\sim N(10,12)$ and $Y\sim N(5,8)$. Find the normal distribution of

$4X+2Y$
$3X-4Y$
${{X}_{1}}+{{X}_{2}}+{{X}_{3}}-{{Y}_{1}}-{{Y}_{2}}-{{Y}_{3}}-{{Y}_{4}}$

Modelling assumption for the indexed variables: $X_1,X_2,X_3$ are independent copies of $X$, $Y_1,\ldots,Y_4$ are independent copies of $Y$, and all seven variables are mutually independent.

Accepted Answer

(i) $4X+2Y\sim N(50,224)$ (ii) $3X-4Y\sim N(10,236)$ (iii) $X_1+X_2+X_3-Y_1-Y_2-Y_3-Y_4\sim N(10,68)$

Linear Combinations of Normal Random Variables

Linear Combinations of Normal Random Variables

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