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 ?

  • 3
    $\begingroup$ 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. $\endgroup$ – André Nicolas Sep 14 '15 at 16:36
  • $\begingroup$ 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. $\endgroup$ – reed20 Sep 14 '15 at 17:02
  • $\begingroup$ The density function of the standard normal (the usual bell-shaped curve) is symmetric about the $y$-axis. So $\Pr(-2\le Z\le -1)=\Pr(1\le Z\le 2)$. $\endgroup$ – 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, $x=2$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.