Asked Oct 24, 2019

If F(x) is the CDF of the standard normal distribution Z~Norm(0,1), show that

(a) For 0 ≤ a < b, P(a < Z < b) = F(-a) - F(-b).

(b) For c > 0, P(-c < Z < c) = 2F(c) - 1.


Expert Answer

Step 1


From the given information, F(X) is the CDF of the standard normal distribution.


Image Transcriptionclose

Р(а<Z <b) -Р(Z <b)-Р(Z<а) 31-P(Z <-b) -[1-P(Z <-а)] 31-Р(Z<-b)-1+P(Z < -а) 3D Р(Z <-а)-Р(Z<-b) —F(-а)-F(-b) P(Z<) F()]

Step 2

For 0≤a<b, P(a<Z<b)=F(-a)-F(-b).

Step 3



Image Transcriptionclose

P(-c<Z<c)= P(Z<c)-P(Z<-c) P(Z <e)-[1-P(Z <e)] [Sin ce, P(Z <-2)=1-P(Z<-2] =P(Z <c)-1+P(Z <c) 2P(Z <c)- 2F (c)-1 P(Z <)=F()]


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