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Using Continuous Uniform MGF to find E(X)

  • #1
210
1

Homework Statement





Homework Equations





The Attempt at a Solution



Continuous Uniform MGF is [itex]M_{x}(z) = E(e^zx) = \frac{e^{zb} - e^{za}}{zb - za}[/itex]

[itex]\frac{d}{dz}M_{x}(z) = E(X)[/itex]

Using the Product Rule

[itex]\ U = e^{bz} - e^{az}[/itex]

[itex]\ V = (zb - za)^{-1}[/itex]

[itex]\ U' = be^{bz} - ae^{az}[/itex]

[itex]\ V' = -1(zb - za)^{-2}(b - a)[/itex]

[itex]\frac{dM}{dz} = UV' + VU'[/itex]

[itex]\frac{dM}{dz} = (e^{bz} - e^{az})(-1(zb - za)^{-2}(b - a)) + ((zb - za)^{-1})(be^{bz} - ae^{az})[/itex] evaluated at z = 1


[itex]\ (e^{b}-e^{a})(-1)(b - a)^{-2}(b - a) + (b - a)^{-1}(be^{b} - ae^{a})[/itex]


[itex]\frac{e^{a} - e^{b} + be^{b} - ae^{a} }{b - a}[/itex]


The answer is [itex]\frac{b + a}{2}[/itex]


I'd really appreciate it if someone could tell me where i'm going wrong. Thanks.
 

Answers and Replies

  • #2
329
34
You didn't define your terms. What is MGF and what is E(X). These may be standard, but I don't know them.
 
  • #3
210
1
Moment Generating Function and Expected Value of the Continuous random variable X
 
  • #4
Ray Vickson
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Homework Statement





Homework Equations





The Attempt at a Solution



Continuous Uniform MGF is [itex]M_{x}(z) = E(e^zx) = \frac{e^{zb} - e^{za}}{zb - za}[/itex]

[itex]\frac{d}{dz}M_{x}(z) = E(X)[/itex]

Using the Product Rule

[itex]\ U = e^{bz} - e^{az}[/itex]

[itex]\ V = (zb - za)^{-1}[/itex]

[itex]\ U' = be^{bz} - ae^{az}[/itex]

[itex]\ V' = -1(zb - za)^{-2}(b - a)[/itex]

[itex]\frac{dM}{dz} = UV' + VU'[/itex]

[itex]\frac{dM}{dz} = (e^{bz} - e^{az})(-1(zb - za)^{-2}(b - a)) + ((zb - za)^{-1})(be^{bz} - ae^{az})[/itex] evaluated at z = 1


[itex]\ (e^{b}-e^{a})(-1)(b - a)^{-2}(b - a) + (b - a)^{-1}(be^{b} - ae^{a})[/itex]


[itex]\frac{e^{a} - e^{b} + be^{b} - ae^{a} }{b - a}[/itex]


The answer is [itex]\frac{b + a}{2}[/itex]


I'd really appreciate it if someone could tell me where i'm going wrong. Thanks.
You need to evaluate M'(z) at z = 0, not at z = 1! Of course, you need to worry about the fact that your formula for M(z) does not apply right at z = 0, so you need to look at limits as z → 0, and use l'Hospital's rule, for example. BTW: the formula you wrote for EX is wrong; it should be
[tex] EX = \left. \frac{d M(z)}{dz} \right|_{z = 0} = M'(0).[/tex] For ##z \neq 0## the derivative of M does not have any particular relation to EX.

However, that would be doing it the hard way. From
[tex] M_X(z) = E e^{zX} = 1 + z \,EX + \frac{z^2}{2} E X^2 + \cdots, [/tex]
we see that we can get the moments of X from the moment-generating-function---that's why it is so named---so all you need is the series expansion of M(z) around z = 0 (that is, the Maclaurin series). That is most easily obtained by just getting the series expansion of the numerator, then dividing by z.
 
Last edited:

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