Solution
"=\\int exp(tx) \\theta exp(-\\theta x) dx"
"=\\theta\\int exp(-x(\\theta - t)) dx"
Integrating the above function from 0 to infinity yields:
Which is the mgf of f(x)
Answer: M(t)=(θ)/(θ−t)
1st moment is obtained by differentiating M(t) with respect to t and setting t to 0:
By division rule, we obtain:
Setting t=0 yields:
"\\theta \/ (\\theta)^2 = 1\/ \\theta"Which is the 1st moment
2nd moment is obtained by finding the second derivative of the mgf with respect to t and setting t to 0:
By division rule, we obtain:
as the second derivative.
Setting t to 0 yields:
Which is the second moment
The 3rd moment is similarly obtained by getting the third derivative of the mgf with respect to t and setting t to 0:
By division rule, we obtain
as the third derivative.
Setting t to 0 yields :
Which is the third moment of f(x)
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