IB Mathematics Applications & Interpretation Question Type 1: Finding the Distribution of a Linear Combination of Independent Normal Variables Exercises

Let $X,Y$ and $Z$ be three independent normal random variables such that $X\sim \text{N}(3,5)$, $Y\sim \text{N}(4,1)$ and $Z\sim \text{N}(0,1)$.

Define $T=2X-3Y+Z$.

The independent random variables $X$, $Y$, and $W$ are normally distributed such that $X\sim N(1,2)$, $Y\sim N(3,2)$, and $W\sim N(1,4)$.

Let $T=2X-3Y+W$.

The random variables $X,Y$, and $Z$ are independent. Let $U=X+Y+Z$. It is given that $U$ follows a normal distribution with a mean of $7$ and a variance of $7$.

Let $X$ and $Y$ be independent normal random variables such that $X\sim N(3,5)$ and $Y\sim N(4,1)$.

The continuous random variable $W$ follows a normal distribution with mean $20$ and variance $85$, such that $W\sim N(20,85)$.

Let $X$ and $Z$ be independent normal random variables such that $X\sim N(3,5)$ and $Z\sim N(0,1)$. Let $M$ be the linear combination $M=3X+Z$.

Let $X,Y$, and $Z$ be independent normal random variables such that $X\sim N(3,5)$, $Y\sim N(4,1)$, and $Z\sim N(0,1)$.

Let $X$ and $Z$ be two independent random variables such that $X\sim N(3,5)$ and $Z\sim N(0,1)$.

Let $X$ and $Y$ be independent normal random variables such that $X\sim N(2,2)$ and $Y\sim N(3,4)$.

Topic: Normal Distribution Inverse Normal Calculations