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Show that if Y1 has a χ2 distribution with ν1 degrees of freedom and Y2 has...

Show that if Y1 has a χ2 distribution with ν1 degrees of freedom and Y2 has a χ2 distribution with ν2 degrees of freedom, then U = Y1 + Y2 has a χ2 distribution with ν1 + ν2 degrees of freedom, provided that Y1 and Y2 are independent.

The moment-generating function for Y1 is mY1(t) = ________   and the moment-generating function for Y2 is mY2(t) = _______. Since Y1 and Y2 are independent, the moment-generating function for U is mU(t) = _________, which is the moment-generating function for a χ2 distribution with ν1 + ν2 degrees of freedom.

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