Question: Let \(Y_{1}\) and \(Y_{2}\) be independent random variables with moment-generating functions \({{m}_{{{Y}_{1}}}}\left( t \right)\) and \(m_{Y_{2}}(t)\), respectively. If \(a_{1}\) and \(a_{2}\) are constants, and \(U=a_{1} Y_{1}+a_{2} Y_{2}\) moment-generating function for \(U\) is \(m_{U}(t)=m_{Y_{1}}\left(a_{1} t\right) \times m_{Y_{2}}\left(a_{2} t\right)\).

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