Probability with Independent Normal Variables

Question

Let X and Y be independent standard normal variables. Find:

a) P(3X + 2Y > 5)

Answer 1

Here are some hints that should go a long way towards answers.

(a) In your course, you may have learned that if $X$ and $Y$ are independent normal with means $p$, $q$ and variances $s^2$, $t^2$, then $aX+bY$ is normal mean $ap+bq$, variance $a^2s^2+b^2t^2$. So $3X+2Y$ has mean $0$, variance $13$.

(b) Our probability is $1$ minus the probability that they are both $\ge 1$. What is the probability that $X \ge 1$?

(c) The probability that the absolute value of $\min(X,Y)$ is less than $1$ is the probability that $(X < 1) \cup (Y<1)$ minus the probability that $(X \le -1) \cup (Y \le -1)$.

(d) This is the probability that $|X-Y|<1$. Let $W=X-Y$ and look at the remark for part (a).