-Several sufficient conditions are formulated for the uni-and bimodality of a mixture of two Gaussian distributions with equal variances σ 2 and different expectation values µ i , i = 1, 2. An equation governing all the degenerate critical inflection points for the probability density f ( x ) of the mixture is derived by a statistical method. This equation describes the boundary of the uni-and bimodality domains of f ( x ).
-Several theorems on sufficient unimodality conditions are formulated for a sum of k normal distributions with the same variance and with different mean values µ i , i = 1, …, k , 2 ≤ k < ∞ , taken with their a priori probabilities π i . On the basis of these theorems, estimates for the lower and upper bounds for the mode numbers m are obtained for k ≥ 3 in the case when the mixture contains k * components, 2 ≤ k * < k , satisfying the unimodality conditions.
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