Applied Statistics and Probability for Engineers

5-10

This last summation is the binomial expansion of [pet(1p)]n, so

Taking the first and second derivatives, we obtain

and

If we set t0 in , we obtain

which is the mean of the binomial random variable X. Now if we set t0 in

Therefore, the variance of the binomial random variable is

EXAMPLE S5-6 Find the moment generating function of the normal random variable and use it to show that

the mean and variance of this random variable are and ^2 , respectively.

The moment generating function is

If we complete the square in the exponent, we have

and then

et

(^2) t (^2
2)



   

1

 12 

e^11

223 x^1 t

(^2242)  2
dx

MX 1 t 2  

   

1

 12 

e^53 x^1 t

(^2242)  2 t (^2) t (^2)  (^46
12)  (^22)
dx
x^2  21 t^22 x^2  3 x 1 t^2242  2 t^2 t^2 ^4

 

   

1

 12 

e^3 x

(^2)  21 t (^22) x (^24
12)  (^22)
dx

MX 1 t 2  

   

etx

1

 12 

e^1 x^2

(^2
12)  (^22)
dx
^2 ¿ 2 ^2 np 11 pnp 2  1 np 22 npnp^2 np 11 p 2
MX– 1 t (^20) t 0 ¿ 2 np 11 pnp 2
M–X 1 t 2 ,
MX¿ 1 t (^20) t 0 ¿ 1 np
MX¿ 1 t 2
MX– 1 t 2 
d^2 MX 1 t 2
dt^2
npet 11 pnpet 231 p 1 et 124 n^2
MX¿ 1 t 2 
dMX 1 t 2
dt
npet 31 p 1 et 124 n^1
MX 1 t 2  3 pet 11 p 24 n
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