= Solution
The <messenger RNA> has <steady state> $X_{R,*}=\beta_R/\alpha_R$. Since <biological translation> produces <proteins> at rate $rX_R$, setting the <protein degradation> rate equal to this production rate gives
$$
\boxed{X_{P,*}=\frac{r\beta_R}{\alpha_P\alpha_R}=\frac{b\beta_R}{\alpha_P},\qquad b=\frac r{\alpha_R}.}
$$
The <translational burst size> is the expected number of <proteins> made during one <messenger RNA> lifetime, whose <expected value> is $1/\alpha_R$. It is dimensionless, rather than a production rate. With independent <Poisson process> translation and an <exponential distribution> for the lifetime, the resulting count has a <geometric distribution> on nonnegative integers with <expected value> $b$ and <variance> $b(1+b)$.
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