Huang, Yu-RuIn this thesis, we consider the quadratic family f_t(x)=tx(1-x), and the set of parameter values t for which f_t has an absolutely continuous invariant measures (a.c.i.m.). It was proven by Jakobson that the set of parameter values t for which f_t has an a.c.i.m. has positive Lebesgue measure. Most of the known results about the existence and the measure of parameter values with a.c.i.m. concern a small neighborhood of the Chebyshev parameter value t=4. Differently from previous works, we consider an interval of parameter not adjacent to t=4, and give a lower bound for the measure of the set of parameter values t for which f_t has an a.c.i.m. in that interval.DissertationMathematics