= Solution
Define the one-dimensional thermal widths $\sigma_p=\sqrt{k_{\rm B}T_p/m_e}$ and $\sigma_b=\sqrt{k_{\rm B}T_b/m_e}$. Normalize each <velocity distribution> to its electron number density:
$$
f_p(v)=\frac{n_p}{\sqrt{2\pi}\sigma_p}e^{-v^2/(2\sigma_p^2)},\qquad
f_b(v)=\frac{n_b}{\sqrt{2\pi}\sigma_b}e^{-(v-V)^2/(2\sigma_b^2)}.
$$
The total is $f=f_p+f_b$. It consists of a broad peak centred at zero and a much narrower displaced beam near $V$, with $V\gg\sigma_p\gg\sigma_b$. Their areas, rather than their peak heights, are $n_p$ and $n_b$. A weak narrow beam can make a positive-slope region on its low-velocity side even while contributing little to the total density.
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012-s16-beam-distribution.png]
{title=Broad plasma Maxwellian and a narrow displaced electron beam}
{description=Illustrative normalized widths and density ratio are used to display the beam on the background tail. The marked V−sigma_b is the point of maximum positive beam slope, not a universal exact marginal-mode velocity.}
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