Let X1 , . . ., Xn denote a sequence of iid Bernoulli (p) random variables and let K = X1 + ∙ ∙ ∙ + Xn. In addition, let M denote a binomial (n,) random variable, independent of X1, ∙ ∙ ∙ + Xn. Do the random variables U = X1 + ∙ ∙ ∙ + XK and V = X1 + ∙ ∙ ∙ + XM have the same expected value? Be cvareful: U is not an ordinary random sum of random variables.