 Solution

ID: past-exam-of-the-systems-biology-course-of-the-university-of-cambridge/2024/paper-4/section-i/1-2/solution

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.

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