Past exam of the systems biology course of the University of Cambridge / 2024 / Paper 4 / Section I / 1.2

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- Solution 1.2

Solution

The probability density function is

f (x) = j = 0 \sum 1 \frac{π _{j}}{σ _{j} 2 π} exp [- \frac{( x - μ _{j} ) ^{2}}{2 σ _{j}^{2}}] .

Every term is nonnegative because its weight is nonnegative, its scale is positive and its exponential is positive. By linearity of the integral and normalization of each normal distribution,

\int_{- \infty}^{\infty} f (x) d x = \sum_{j = 0}^{1} π_{j} \int_{- \infty}^{\infty} ϕ_{μ_{j}, σ_{j}} (x) d x = π_{0} + π_{1} = 1.

Thus

f

is nonnegative and integrates to one. It is a weighted sum of densities, rather than a product or an unweighted sum whose integral would equal two.

Figure 1.
Illustrative Gaussian mixtures with shared component distributions and different modification fractions
. Details.

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