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Unformatted text preview: To do that, start with ∞ xn =
n=0 Diﬀerentiating, ∞ nxn−1 =
n=1 1
.
1−x
1
.
(1 − x)2 Letting x = 1 − p, we obtain
EX = 1
1
p= .
(1 − (1 − p))2
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 p = 1/2, so
the expected number of children is 2; one of them is a boy, so the expected
number of girls is 1.
The quantity E X k is called the k th moment of X , and the ﬁrst moment
is also called the mean.
If we roll a die,
E X 2 = 1 · 1 + 4 · 1 + · · · + 36 ·
6
6 1
6 = 91
.
6 Make the observation that if X is equal to a with probability one, then
E X = a.
Proposition 3.4
E (X + a) = E X + a
and
E (aX ) = aE X. 8 ...
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan

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