Answer to Question #187044 in Statistics and Probability for sphoorthi

Question #187044

Suppose that X

X has pdf given by

f(x)={x=1 if 1<x<2

3-x if 2<x<3

0 elsewhere

the MGF of X is


1
Expert's answer
2021-05-02T17:57:50-0400

The given pdf is-

f(x)=x, if 1<x<2

3-x, if 2<x<3

0, elsewhere


The moment generating function is-

"M=E(e^{tx})"

"=\\int_0^{\\infty}f(x) e^{tx}dx"


"=\\int_1^2xe^{tx}dx+\\int_2^3(3-x)e^{tx}dx"


"=x\\dfrac{e^{tx}}{t}|_1^2-\\dfrac{e^{tx}}{t^2}|_1^2+3\\dfrac{e^{tx}}{t}|_2^3-(x\\dfrac{e^{tx}}{t}|_2^3-\\dfrac{e^{tx}}{t^2}|_2^3)"


"=2\\dfrac{e^{2t}}{t}-\\dfrac{e^{t}}{1}-(\\dfrac{e^2t-e^t}{t^2})+3(\\dfrac{e^{3t}}{t}-\\dfrac{e^{2t}}{2})-[3\\dfrac{e^{3t}}{t}-2\\dfrac{e^{2t}}{t}-\\dfrac{e^{3t}}{t^2}+\\dfrac{e^{2t}}{t^2}]"


"=-\\dfrac{3e^{2t}}{2}+\\dfrac{4e^{2t}}{t}-e^{-t}+\\dfrac{e^t}{t^2}-\\dfrac{2e^{2t}}{t^2}+\\dfrac{e^{3t}}{t^2}"


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