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# S2103_MA340_Q4_sol.pdf

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Wilfrid Laurier University

Mathematics

MA340

Adam Metzler

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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