elemprob-fall2010-page9

# elemprob-fall2010-page9 - Proof. The ﬁrst one is a...

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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. 9 ...
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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.

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