Applied Statistics and Probability for Engineers

4-1

Mean and Variance of the Normal Distribution (CD Only)

In the derivations below, the mean and variance of a normal random variable are shown

to be and ^2 , respectively. The mean of xis

By making the change of variable , the integral becomes

The first integral in the expression above equals 1 because is a probability density

function and the second integral is found to be 0 by either formally making the change of vari-

able uy^2 2 or noticing the symmetry of the integrand about y0. Therefore, E(X) .

The variance of Xis

By making the change of variable , the integral becomes

Upon integrating by parts with and V(X) is found to be.

4-8 CONTINUITY CORRECTIONS TO IMPROVE

THE APPROXIMATION

From Fig. 4-19 it can be seen that a probability such as P(3 X 7) is better approximated

by the area under the normal curve from 2.5 to 7.5. This observation provides a method to im-

prove the approximation of binomial probabilities. Because a continuous normal distribution

is used to approximate a discrete binomial distribution, the modification is referred to as a

continuity correction.

dvy ^2

ey

(^2)  2
22 
uy dy,

V 1 X 2 ^2 

   

y^2

ey

(^2)  2
22 
dy
y 1 x (^2) 

V 1 X 2  

   

1 x 22

e^1 x^2

(^2)  2  2
22 
dx
ey
(^2)  2
22 

E 1 X 2 

   

ey

(^2)  2
22 

dy



   

y

ey

(^2)  2
22 
dy
y 1 x (^2) 

E 1 X 2  

   

x

e^1 x^2

(^2)  2  2
22 
dx
If Xis a binomial random variable with parameters nand p, and if x 0, 1, 2,p , n,
the continuity correctionto improve approximations obtained from the normal dis-
tribution is
and
P 1 xX 2 P 1 x0.5X 2 P °
x0.5np
2 np 11 p 2
Z¢
P 1 Xx 2 P 1 Xx 0.5 2 P °Z
x 0.5np
2 np 11 p 2
¢
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