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Unformatted text preview: e is 91/6 49/4 = 70/24. Taking the square root we see that
the standard deviation is 1.71. The three possible deviations, in the sense of
X E X , are 0.5, 1.5, and 2.5 with probability 1/3 each, so 1.71 is indeed a
reasonable approximation for the typical deviation from the mean.
Example A.19. Geometric distribution. Starting with the sum of the
geometric series
1
X
(1 ✓) 1 =
✓n
n=0 and then di↵erentiating twice and discarding terms that are 0, gives
(1 ✓) 2 = 1
X n✓ n 1 and 2(1 ✓) 3 = n=1 Using these with ✓ = 1
EN =
EN (N 1) = 1
X n=1
1
X 1
X n(n 2 n=2 p, we see that
n(1 p)n n(n 1)(1 1 p = p/p2 =
p)n 1 1
p p = 2p 3 (1 p)p = n=2 and hence 1)✓n var (N ) = EN (N
= 1) + EN 2(1 p)
p
+2
p2
p 2(1 p)
p2 (EN )2 1
(1 p)
=
p2
p2 The deﬁnition of expected value generalizes in the usual way to continuous
random variables. We replace the probability function by the density function
and the sum by an integral
Z
Eh(X ) = h(x)fX (x) dx
(A.19) 217 A.3. EXPECTED VALUE, MOMENTS Example A.20. Uniform distribution...
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 Spring '10
 DURRETT
 The Land

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