To do that, start with
∞
n
=0
x
n
=
1
1

x
.
Differentiating,
∞
n
=1
nx
n

1
=
1
(1

x
)
2
.
Letting
x
= 1

p
, we obtain
E
X
=
1
(1

(1

p
))
2
p
=
1
p
.
This is reasonable: the number of tosses to get a head should be about
1
/p
.
For a “reallife” example, in China, parents are allowed to have only one
child.
However, in rural areas, if the child is a girl, they are allowed to
have a second child. Suppose the rule were that parents could have children
until they had a boy. How many girls would a family have on average? The
number of children per family has a geometric distribution with
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 ansan
 Probability theory, th moment, ﬁrst moment

Click to edit the document details