Let [math]X[/math] be a continuous random variable with values in [math][\,0,\infty)[/math] and density [math]f_X[/math]. Find the moment generating functions for [math]X[/math] if

• [math]f_X(x) = 2e^{-2x}[/math].
• [math]f_X(x) = e^{-2x} + (1/2)e^{-x}[/math].
• [math]f_X(x) = 4xe^{-2x}[/math].
• [math]f_X(x) = \lambda(\lambda x)^{n - 1} e^{-\lambda x}/(n - 1)![/math].