Let X and Y be independent random variables with moment-generating function MX(t) and MY (t) respectively. If a and b are constants and U = aX +bY , show that the moment-generating function of U is MU (t) = MX(at).MY (bt).
"M_X(t) = E(e^{tX}) \\\\\n\nM_Y(t) = E(e^{tY})"
X and Y are independent
"U = aX +bY \\\\\n\nM_U(t) = E(e^{tU}) = E[e^{t(aX+bY)}] \\\\\n\n = E[e^{(at)X} \\times e^{(bt)Y}] \\\\\n\n= E[e^{(at)X}] \\times E[e^{(bt)Y}] \\\\\n\nM_U(t) = M_X(at) \\times M_Y(bt)"
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