Máximo da Distribuição Multinomial

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The Multinominal Distribution and Stirling's Approximation 173 Since both numerator and denominator are undefined as x 0, we can apply l'Hôpital's rule: = = E-9. Prove that the maximum value of W(N, N₁) = N₁)!N₁ ! is given by N₁ = N/2. (Hint: Treat N₁ as a continuous variable.) Let N₁ = x, with x being a continuous variable. Then In = In N! In(N We are looking for the maximum value of W, so, at this value, (dW/dx) = 0. Also, we know that W 1 dW dx = dx W) At the maximum value of W, then, W)/dx = 0. We can express In W using Stirling's approximation (Equation E.7): In W = In N! In(N - = N [(N x) (N x)] (x x) The derivative of In W is then Setting W)/dx] = 0 gives = = N II Recall that N₁ = x, and that N₂ = N - The maximum value of W then occurs when N₁ = N/2. E-10. Prove that the maximum value of W(N₁, N₂, N,) in Equation E.5 is given by Given: N!