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Let $T=10\,\mathrm{Gyr}$ and let $t$ denote present <stellar age>, rather than time measured from the galaxy's birth. With a fixed <initial mass function>, constant <star formation rate> also gives a constant number of births per unit time, as in the <birth-age distribution under constant star formation>. Counting the supplied <stellar evolution> model's remnants as part of the population, ages have a <uniform distribution> on $[0,T]$. Integrating the constant birth rate therefore gives
$$
\boxed{Y(t)=\frac tT=0.1\frac{t}{\mathrm{Gyr}}\quad(0\le t\le T).}
$$
Outside this interval the cumulative fraction is zero or one as appropriate.
Write $m=M/M_\odot$ for the birth-mass parameter and $M_{\min}=0.2M_\odot$. The normalization cancels when integrating the <initial mass function>:
$$
X(M)=\frac{\int_M^\infty kM_\odot^3M'^{-3}\,dM'}{\int_{M_{\min}}^\infty kM_\odot^3M'^{-3}\,dM'}=\frac{M_{\min}^2}{M^2}.
$$
Thus \b[the mass-tail fraction] is
$$
\boxed{X(M)=\frac{0.04}{m^2}\quad(m>0.2).}
$$
It is one at and below the lower cutoff. The normalization constant has no effect on any number fraction.
For <evolutionary-state selection by stellar lifetimes>, a <red giant> has completed its <main sequence> lifetime but not its giant lifetime. Its <stellar lifetime regions in a mass-age diagram> are bounded by
$$
\boxed{\frac{8\,\mathrm{Gyr}}{m^2}<t<\frac{10\,\mathrm{Gyr}}{m^2},\qquad 0<t<T,\qquad m\ge0.2.}
$$
The lower boundary meets $t=T$ at $m=\sqrt{0.8}$ and the upper boundary meets it at $m=1$. A <white dwarf> lies above the upper lifetime curve, within the same age interval.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-322-population-regions.png]
{title=Red-giant and white-dwarf regions in the birth-mass–age plane and their triangular images in uniform mass-tail–age coordinates}
The <probability integral transform>, applied to the decreasing mass-tail coordinate, makes $X$ uniform on $(0,1)$: $\Pr(X<x)=x$. The independent birth age gives $Y=t/T$ uniform on $(0,1)$, so number fractions are areas in a unit square. Since $m^{-2}=25X$, the <red giant> region becomes $20X<Y<25X$, a triangle with vertices $(0,0)$, $(1/25,1)$ and $(1/20,1)$. Its area is $\tfrac12(1/20-1/25)=1/200$. The <white dwarf> region $Y>25X$ is a triangle of area $\tfrac12(1/25)=1/50$. Hence \b[the present individual-star fractions] are
$$
\boxed{\Pr(G)=0.005=0.5\%,\qquad\Pr(W)=0.02=2\%.}
$$
The rest are on the <main sequence> under the given toy lifetime model.
For the <binary stars>, draw two independent mass-tail coordinates $X_1,X_2$, but only one shared age coordinate $Y$: components born together are coeval. The <coeval binary population> therefore occupies a uniform unit cube $(X_1,X_2,Y)$. At a fixed age $Y=y$, the state <probabilities> for either component are
$$
g(y)=\Pr(G\mid y)=\frac y{100},\qquad w(y)=\Pr(W\mid y)=\frac y{25}=4g(y),\qquad s(y)=1-\frac y{20}.
$$
Their evolutionary states have <conditional independence> given $y$. They generally do not have unconditional independence, because sharing an age correlates their states. The binary fractions below are slice areas averaged over $0<y<1$.
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