Computing a Bimodal Gaussian Function Normalized to Unit Area

A bimodal Gaussian function is a probability density function formed by superimposing two independent Gaussian distributions. To compute a bimodal Gaussian function normalized to unit area, the total area under the curve must equal 1, which can be achieved by assigning appropriate weights to each Gaussian component. In code implementations, this involves calculating the combined integral and applying normalization factors.

For normalized bimodal Gaussian functions, linear combination is the standard construction method. Assuming two Gaussian peaks with means μ₁ and μ₂ and standard deviations σ₁ and σ₂ respectively, the normalization constraint requires the sum of coefficients for both Gaussian components to equal 1, ensuring the integral of the entire function remains unity. The mathematical formulation can be implemented using Gaussian probability density functions with weighted summation.

When constructing bimodal functions, correctly setting the relative weights of the two Gaussian components is crucial. A common approach assigns 0.5 weight to each peak, making both peaks contribute equally to the overall function. Alternatively, adjusting the weight ratio alters the relative significance of the peaks. Programmatically, this involves defining weight parameters that sum to 1.0 and scaling each Gaussian component accordingly.

In numerical computations, we typically sample points around both peaks and perform weighted summation. The implementation requires selecting a sufficiently large interval range to fully cover both peaks' influence regions. For larger standard deviation values, wider sampling ranges may be necessary. Code implementations often use vectorized operations for efficient computation across the sampling domain.

This bimodal structure is prevalent in signal processing, statistics, and pattern recognition, particularly in mixture models containing two primary components or states. Proper normalization ensures the function maintains valid probabilistic interpretability, which is essential for statistical inference and model fitting algorithms.