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# S2103_MA340_Q4_sol.pdf

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School
Wilfrid Laurier University
Department
Mathematics
Course
MA340
Professor
Semester
Spring

Description
MA340 Quiz: July 25-26, 2013 ||| Solutions 1. Let X have probability mass function f(x) = (:3)(:7)1; x = 1;2;3;::: : Derive the moment generating function of X. Be sure to state (and justify) its domain. t Hint: At some point you’re going to want to factor out (:3)e from each term in a series. tX M(t) = E[e ] 1 X tx x▯1 = e (:3)(:7) x=1 1 X = et(x▯1+(:3)(:7)▯1 x=1 X1 = et(x▯1)(:3)(:7)1 x=1 X1 = et(x▯e (:3)(:7)1 x=1 X1 t t x▯1 x▯1 = (:3e ) (e ) (:7) x=1 1 tX t x▯1 = (:3e ) (:7e ) : x=1 t The series above is a geometric series and will converge if and only if j:7e j < 1, which is equivalent to t < ln(10=7) = 0:356:::. Provided t lies in the appropriate range the sum of the series isherefore 1▯:7e :3et M(t) = ; t < ln(10=7) : 1 ▯ :7e 2. Suppose that X has moment generating function (:9)e M Xt) = ; t < ln(10) : 1 ▯ (:1)e (a) Use MX (or a suitable transformation Xf M ) to ▯nd the mean and variance of X. We have t M (t) = :9e ; X (1 ▯ :1e)2 00 (:9e )(1 + :1e ) M Xt) = : (1 ▯ :1e)3 Therefore :9 E[X] = 2 (1 ▯ :1) :9 = (:9) 1 = :9 10 = 9 = 1:11::: ; and 2 00 E[X ] = M X0) = (:9)(1 + :1) (1 ▯ :1) (:9)(1:1) = 3 (:9) 1:1 = (:9) = 110 81 = 1:358::: : So the mean of X is 10=9 and the variance is Var(X) = E[X ] ▯ (E[X])2 110 100 = 81 ▯ 81 10 = 81 = 0:123::: (b) Use MX (or a suitable transformation Xf M ) to ▯nd the mean and variance of Y = e . The mean of Y is X E[Y ] = E[e ] 1▯X = E[e ] = M X1) :9e = 1 ▯ :1e = 3:359::: ; and the variance is 2 2 Var(Y ) = E[Y ] ▯ (E[Y ]) X 2 2 = E[(e ) ] ▯ (MX(1)) 2▯X 2 = E[e ] ▯ (MX(1)) 2 = M X2) ▯ (M X1)) 2 ▯ ▯ 2 = :9e ▯ :9e 1 ▯ :1e 1 ▯ :1e = 25:47::: ▯ (3:35:::) = 14:18::: : (c) Use M (or a suitable transformation of M ) to ▯nd the moment generating function of Z = 4X ▯ 1. X X M Zt) = E[exp(tZ)] = E[exp(t[4X ▯ 1])] = E[exp(4tX)exp(▯t)] ▯t 4tX = e E[e ] ▯t = e M X4t) :9et = e▯t 1 ▯ :1et 3t :9e = 1 ▯ :1et: Note that M (4t) is only well-de▯ned if 4t < ln(10), or t < ln(10)=4. Therefore X 3t M (t) = :9e ; t < ln(10)=4 :
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