Figure 2. Accuracy of our integral representations for the Telegraph and negative binomial distribution.

(A) For each of the results in (3 - 5), we compare the (fixed-parameter) Telegraph and negative binomial distributions with their respective compound representations for two different sets of parameter values. The top panel (pink) shows comparisons for (3), with parameter values (left) $\lambda =2$, ${\mu }^{\prime }=12$, $K^{\prime }=100$, $\mu =3$, and ${\displaystyle K\sim {\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}}_{K^{\mathrm{\prime }}}(5,9)}$, and (right) $\lambda =1$, ${\mu }^{\prime }=20$, $K^{\prime }=100$, $\mu =2$ and ${\displaystyle K\sim {\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}}_{K^{\mathrm{\prime }}}(3,18)}$. The middle panel (green) gives comparisons for (4), with parameter values (left) $\lambda =10$, $\beta =2$, $\mu =2$ and ${\displaystyle K\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(12,2)}$ and (right) $\lambda =1$, $\beta =1$, $\mu =2$ and ${\displaystyle K\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(3,1)}$. The bottom panel (coral) gives comparisons for (5). The parameter values (left) are ${\lambda }^{\prime }=10$, $\lambda =15$ and $c=2$ and (right) are ${\lambda }^{\prime }=2$, $\lambda =5$ and $c=3$. (B) The top figure compares a ${\displaystyle \mathrm{T}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}(2,4,60)}$ distribution with samples from a compound Telegraph distribution with normal noise ${\displaystyle \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}(37,10)}$ on the transcription rate parameter. The middle figure compares a ${\displaystyle \mathrm{N}\mathrm{e}\mathrm{g}\mathrm{B}\mathrm{i}\mathrm{n}(5,0.5)}$ with samples from a compound Telegraph distribution with normal noise ${\displaystyle \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}(5.5,2.3)}$ on the transcription rate parameter. The bottom figure compares a ${\displaystyle \mathrm{N}\mathrm{e}\mathrm{g}\mathrm{B}\mathrm{i}\mathrm{n}(5,1)}$ distribution with samples from a compound negative binomial distribution with normal noise ${\displaystyle \mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}(2.3,0.6)}$ on the burst intensity parameter.