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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,

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    Maybe assume that we have maximum where some $k_i$ is "too small" (other case, "too big"). Then since the sum of the $k_i$ is fixed, as is the sum of the probabilities, some $k_j$ is "too big". See what happens when we increase $k_i$ by $1$, decrease $k_j$ by $1$, leaving all the rest alone. A ratio calculation like in the $n=2$ case should work.2011-07-25

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