Begin2.DVI

11-11. (Binomial Distribution)

The discrete binomial distribution is given by f(x) =

{(n

x

)

px(1 −p)n−x, x = 0, 1 , 2 , 3 ,...

0 , otherwise

If q= 1 −pshow that

(a) (p+q)n=

∑n

x=0

f(x) = 1 , (b)

∑n

x=1

(n− 1

x− 1

)

px−^1 qn−x= (p+q)n−^1 , (c)x

(n

x

)

=n

(n− 1

x− 1

)

(d) Use parts (b) and (c) to show the mean of the binomial distribution is given by

μ=E[x] =

∑n

x=0

xf (x) =

∑n

x=0

x

(

n

x

)

pxqn−x=np

11-12.

(a) Show that s^2 =^1

N− 1

∑N

j=1

(xj−x ̄)^2 can be written in the shortcut form

s^2 =

1

N(N−1)





N

∑N

j=1

x^2 j−





∑N

j=1

xj





2 



(b) Illustrate the use of the above two formulas by completing the table below and

evaluating s^2 by two different methods.

x x^2 x− ̄x (x−x ̄)^2

6

3

8

5

2

∑^5

j=1

xj=

∑^5

j=1

x^2 =

∑^5

j=1

(x− ̄x)^2 =

x ̄=





∑^5

j=1

xj





2

=