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Chapter 5: Q. 5.9 (page 215)

Show that Zis a standard normal random variable; then, for,x>0

  1. P{Z>x}=P{Z<−x}
  2. P{|Z|>x}=2P{Z>x}
  3. P{|Z|<x}=2P{Z<x}−1

Short Answer

Expert verified
  1. The value P{Z>x}=P{Z<-x}is proved.
  2. The valueP{|Z|>x}=2P{Z>x}is proved.
  3. The value P{|Z|<x}=2P{Z>x}-1is proved.

Step by step solution

01

Determine the random variable (part a)

Define Z~N(0,1)

We have that P(Z>x)=P(-Z<-x)=P(Z<-x)

where the first equation hold since we have multiplied the inequality with -1and the second equality holds since Zand -Zhave the same distribution sinceZis symmetric around zero.

02

Evaluate the values (part b)

We have that

P(|Z|>x)=P((Z>x)∪(Z<-x))=P(Z>x)+P(Z<-x)=2P(Z>x)

Where we have proved the last equality in (part a)

03

Evaluate the value (part c)

We have that

P(|Z|<x)=P(-x<Z<x)=2P(0<Z<x)=2P(Z<x)-12

The second equality holds because of the symmetry of the interval (-x,x)and then we have added interval (-∞,0)to the interval (0,x)and we know thatP(Z<0)=12

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