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