I am reading Albert N Shiryaev's Probability. There is one question from Chapter I §2.
Problem 2: Show that for the multinomial distribution $\{P(A_{n1},\ldots, A_{nr})\}$ the maximum probability is attained at a point $(k_1, \ldots, k_r)$ that satisfies the inequalities $np_i-1< k_i\le(n+r-1)p_i, i=1,\ldots,r$.
The probability of Multinomial distribution can be found from wiki
May I have any hint on how to prove that? If possible, can we have the expression for this maximum possibility? I am thinking of proving it using Lagrange multiplier. However, the Lagrange multiplier may only useful for continuous case?
Thanks,