= Solution
The <probability density function> is
$$
\boxed{f(x)=\sum_{j=0}^1\frac{\pi_j}{\sigma_j\sqrt{2\pi}}\exp\!\left[-\frac{(x-\mu_j)^2}{2\sigma_j^2}\right].}
$$
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>,
$$
\int_{-\infty}^{\infty}f(x)\,dx=\sum_{j=0}^1\pi_j\int_{-\infty}^{\infty}\phi_{\mu_j,\sigma_j}(x)\,dx=\pi_0+\pi_1=1.
$$
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.
\Image[/past-exam-of-the-systems-biology-course-of-the-university-of-cambridge/2024/paper-4-mixture-weights.png]
{title=Illustrative Gaussian mixtures with shared component distributions and different modification fractions}
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