What to wear to a hospital delivery room? If you’re Joan Rivers, and your daughter is about to have a baby, it’s a designer outfit, of course. “I had on a very simple Armani black suit and a Valentino blouse,” Rivers, 67, told PEOPLE. “They kept saying to me, ‘Will you put on scrubs?’ and I thought, ‘Why?’ ” Likewise, the mama of the hour, Melissa, 32, “made sure her bangs didn’t frizz,” says the senior Rivers. “It was a very California birth — everyone looked terrific.” That included the underdressed main attraction: Edgar Cooper Endicott, the 6-lb. 15-oz. son born to Melissa, who cohosts (with her mother) the annual Oscar and Emmy fashion shows on E! Entertainment Television, and her husband of two years, horse trainer and breeder John Endicott, 34. Edgar Cooper “has my big ol’ lips,” adds Melissa, noting that the new arrival (who was nearly named Elvis) “looks just like my dad,” his namesake, the late Edgar Rosenberg.