# Standard Normal Random Variable Question

I am having a bit of trouble with this question here

P {-2 < Z < -1} = P { 1 < Z < x}

The question asks to find the value of X when Z is a standard normal random variable.

How would one approach this question? Do you solve the Z score for the first part first ?

• The best approach is to sketch the density function of the standard normal, and then to use symmetry. The probability that $-2\le Z\le -1$ is the area of a certain region. You could first compute $\Pr(-2\le Z\le -1)$ first, but that is definitely not the best strategy for this problem. – André Nicolas Sep 14 '15 at 16:36
• Im still a bit confused, I can easy sketch the graph here and find probability -2 < Z < -1 but im still frustrated on how I would find x. – reed20 Sep 14 '15 at 17:02
• The density function of the standard normal (the usual bell-shaped curve) is symmetric about the $y$-axis. So $\Pr(-2\le Z\le -1)=\Pr(1\le Z\le 2)$. – André Nicolas Sep 14 '15 at 17:06

Write it in terms of the standard Normal distribution.Use $\Phi(x)=1-\Phi(-x)$.
Hence $$P(-2 < Z < -1)=\Phi(-1)-\Phi(-2)=\Phi(2)-\Phi(1).$$ Hence, $x=2$.