I am struggling to prove the following:

If Z~N $(0,1)$, prove that for the positive k,

$P(|Z|<k)=2-2 \Phi (k)$

I know that $P(|Z|<k)$ can be written as $P(-k<Z<k)$ and that $\Phi (k)$ can be written as $f(z) = \frac{1}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2/(2\sigma^2)}$, but other than that, I am struggling with how to arrive from the LHS to the RHS.

Please advise.Sorry in advance for wrong tags and title labelling.


$P(-k<Z<k)=P(Z<k)-P(z<-k)=P(z<k)-(1-P(z>k))$. Now, using that $f$ is an even function if $Z\sim N(0,1)$m try to calculate $P(z>k)$.


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