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I missed my class that introduced us to moment generating functions, and my notes are missing some pretty essential parts to helping me understand them, so here it is:
Find the moment generating function of [itex]f_X(x)=e^{-x}, x>0[/itex]
[tex]E(X^R)=\int_{-\infty}^{\infty}X^Rf_X(x)dx[/tex]
R must be a natural number I believe.
[tex]m_X(u) = E(e^{uX})[/tex]
I'm unsure if the expectation is applied in the same way, in other words, would this be correct?
[tex]m_X(u) = E(e^{ux})[/tex]
[tex]= \int_{-\infty}^{\infty}e^{uX}f_X(x)dx[/tex]
[tex]= \int_{0}^{\infty}e^{ux}e^{-x}dx[/tex]
[tex]= \int_{0}^{\infty}e^{x(u-1)}dx[/tex]
And I'm not quite sure what the u in this case is either.
Homework Statement
Find the moment generating function of [itex]f_X(x)=e^{-x}, x>0[/itex]
Homework Equations
[tex]E(X^R)=\int_{-\infty}^{\infty}X^Rf_X(x)dx[/tex]
R must be a natural number I believe.
[tex]m_X(u) = E(e^{uX})[/tex]
The Attempt at a Solution
I'm unsure if the expectation is applied in the same way, in other words, would this be correct?
[tex]m_X(u) = E(e^{ux})[/tex]
[tex]= \int_{-\infty}^{\infty}e^{uX}f_X(x)dx[/tex]
[tex]= \int_{0}^{\infty}e^{ux}e^{-x}dx[/tex]
[tex]= \int_{0}^{\infty}e^{x(u-1)}dx[/tex]
And I'm not quite sure what the u in this case is either.