I imagine Trixie being the product of a seedy relationship between Mr. Jack Pot and a Miss Lulamoon (Trixie having "Lulamoon" as a legal surname instead of "Pot"). Miss Lulamoon being a showgirl (or even an out-and-out prostitute), she was probably forced to run away from Las Pegasus after she got pregnant and was forced to raise Trixie alone - Miss Lulamoon is probably now also dead or otherwise not in contact with her daughter because of this.