This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here.
ðɪs iːkwˈɔlᵻɾi dʌznˌɑːt hˈoʊld fɔːɹ ˈʌðɚ pɹˌɑːbəbˈɪlᵻɾi dˌɪstɹɪbjˈuːʃənz, æz ˈɪləstɹˌeɪɾᵻd fɚðə lˈɔɡnˈoːɹməl dˌɪstɹɪbjˈuːʃən hˈɪɹ.