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
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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