Question Type 1: Finding the distribution of a linear combination of independent normal variables Exercises

Let $V = X - Y$. Determine the distribution of $V$.

Given independent random variables $X\sim N(3,5)$, $Y\sim N(4,1)$ and $Z\sim N(0,1)$, find the distribution of $W = 4X + 2Y - Z.$

Let $U = X + Y + Z$. Determine the distribution of $U$.

For $W$ as defined above, compute $P(W>20)$.

Compute $P(X - Y > -1)$.

Define $T=2X - 3Y + Z$. Find the distribution of $T$.

For $U=X+Y+Z$, calculate $P(7<U<9)$.

For $T=2X-3Y+Z$, compute $P(T<0)$.

Find the value of $c$ such that $P(W<c)=0.90$ for $W\sim N(20,85)$.

Define $M=3X + Z$. Find $P(M<10)$.

Find $d$ such that $P(T>d)=0.05$ for $T\sim N(-6,30)$.

Find the 97.5th percentile of $M=3X+Z$, that is, the value $m$ such that $P(M<m)=0.975$.