Question

Z~N (0,1^2) find a) P (Z>2.0) b) P (Z <-2.0) c) P ( -1 <Z <0.5)

Answer 1

$Z\sim N(0,1/2)$ , $\mu =0,\ \sigma ^2=1/2$

We will use the following properties:

$P(X<a)=F(a)=1-F(-a)$

$P(X>a)=1-F(a)=F(-a)$

$P(a<X<b)=F(b)-F(a)$

$F(x)=\Phi (\frac{x-\mu}{\sigma})$

a) $P(Z>2)=1-F(2)=1-\Phi(2\sqrt2)\approx 1-\Phi(2.82)=1-0.9976=0.0024$

b) $P(Z<-2)=F(-2)=1-F(2)=0.0024$

c) $P(-1<Z<0.5)=F(0.5)-F(-1)=\Phi(1/\sqrt 2)-\Phi(-\sqrt 2)\approx\Phi(0.70)- \Phi(-1.41)=0.7580-0.0793\approx 0.68$