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Unformatted text preview: Proof. The ﬁrst one is a special case of E (X1 + X2 ) = E X1 + E X2 with
X2 = a. The second comes from
E (aX ) = (ax)P(X = x) = a xP(X = x) = aE X. The variance of X is deﬁned by
Var X = E (X − E X )2 .
A calculation shows that
Var X = E X 2 − E (2X E X ) + E (E X )2
= E (X 2 ) − 2(E X )E X + (E X )2
= E X 2 − (E X )2 .
So for a die, the variance is
91
7
−
6
2 2 = 105
= 2.9166.
36 The square root of the variance is called the standard deviation.
The variance of a geometric distribution is
1−p
.
p2
To see this, ∞ n(n − 1)(1 − p)n−1 p. E [X (X − 1)] =
n=2 Diﬀerentiating nxn−1 again,
∞ n(n − 1)xn−2 =
n=2 2
.
(1 − x))2 Set x = 1 − p to get
E [X (X − 1)] = 2(1 − p)/p2 .
Some algebra ﬁnishes it.
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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
 Variance

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