Let [math]X[/math] be a continuous random variable with values in [math][\,0,2][/math] and density [math]f_X[/math]. Find the moment generating function [math]g(t)[/math] for [math]X[/math] if

• [math]f_X(x) = 1/2[/math].
• [math]f_X(x) = (1/2)x[/math].
• [math]f_X(x) = 1 - (1/2)x[/math].
• [math]f_X(x) = |1 - x|[/math].
• [math]f_X(x) = (3/8)x^2[/math].

Hint: Use the integral definition, as in example and example.