Let $X_1,...,X_n$ be $n$ independent Bernoulli random variables. (We don't assume that $X_1,...,X_n$ have the same distribution!) Let $Y_1,..., Y_n$ be another $n$ independent Bernoulli random variables. (We don't assume that $Y_1,..., Y_n$ have the same distribution, either!) Let $X = X_1 +....+X_n$ and $Y = Y_1 +....+Y_n$. Suppose that $P(X_i = 1) ge P(Y_i = 1)$ for all $i = 1, 2,...,n$. Does this guarantee that $P(X ge k) ge P(Y ge k)$ for all $k = 1,2,...,n$? If your answer is yes, prove this statement. If your answer is no, give a counterexample.