description = { switch (mode) { case "log-normal": return md`The **log-normal distribution** is parameterized by a *log-mean* ${tex`\mu`} and a *log-variance* ${tex`\sigma^2 > 0`}. It's often used to model phenomena that are affected by many independent, multiplicative factors; to see this, consider the central limit theorem on a log-scale. Such phenomena range from the length of chess games to a user's screen time.

*analytical* mean interarrival: ${tex`~~\,\,\exp\left\{\,\mu + \frac{\sigma^2}{2}\,\right\} \approx ${ Math.round(Math.exp(param[0] + param[1] / 2) * 100) / 100 }`} *empirical* mean interarrival: ${tex`~~~~~~~~~~~~~~~~~~~~~~~~~~\,\,\,~~\approx`} ${tex`${ Math.round(diagram[1] * 100) / 100 }`}

*analytical* mean spread: ${tex`~~~~~~~~~\,\,\,\exp\left\{\mu + \frac{3\sigma^2}{2}\right\} \approx ${ Math.round(Math.exp(param[0] + (3 * param[1]) / 2) * 100) / 100 }`} *empirical* mean spread: ${tex`~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\,\,\,\,~~\,\,\,\approx`} ${tex`${ Math.round(diagram[2] * 100) / 100 }`}

`; break; case "chi-squared": return md`The **chi-squared distribution** is characterized by its *degrees of freedom* ${tex`\nu \in \mathbb{Z}_+`}. Its typical use case is as a pivotal distribution for various statistical tests and not as an interarrival distribution, but let's see how the inspection paradox holds up anyways. This is again a special case of the gamma distribution: ${tex`X \sim \chi^2_\nu \equiv \mathrm{Gamma}(\nu/2, 2)`}.

*analytical* mean interarrival: ${tex`~~~~~~~~~~~~\,\,\nu = ${param}`} *empirical* mean interarrival: ${tex`~~~~~\,\,\,\,\,\,~~~~\,\,\,\,~~\approx`} ${tex`${ Math.round(diagram[1] * 100) / 100 }`}

*analytical* mean spread: ${tex`~~~~~~~~~~~~\,\,\,\nu + 2 = ${ param + 2 }`} *empirical* mean spread: ${tex`~~~~~~~~~~~\,\,~~~~~~~~~~~~~ \approx`} ${tex`${ Math.round(diagram[2] * 100) / 100 }`}

`; case "exponential": return md`The **exponential distribution** is parameterized by a *rate* ${tex`\lambda > 0`}. Recall that a renewal process with exponentially-distributed interarrival times defines a (homogeneous) **Poisson process** with rate ${tex`\lambda`}, having the special property of "memorylessness". This is a special case of the gamma distribution: ${tex`X \sim \mathrm{Exponential}(\lambda) \equiv \mathrm{Gamma}(1, \lambda)`}.

*analytical* mean interarrival: ${tex`~~~~~~~~~~~~~ \frac1{\lambda} \approx ${ Math.round((1 / param) * 100) / 100 }`} *empirical* mean interarrival: ${tex`~~~~~~~~~~~~~~~\,~~\approx`} ${tex`${ Math.round(diagram[1] * 100) / 100 }`}

*analytical* mean spread: ${tex`~~~~~~~~~~~~~~~~~~~~\,\frac{2}{\lambda} \approx ${ Math.round((2 / param) * 100) / 100 }`} *empirical* mean spread: ${tex`~~~~~~~~~~~~~~~~~~~~~~~~\,\,\approx`} ${tex`${ Math.round(diagram[2] * 100) / 100 }`}

`; break; case "Erlang": return md`The **Erlang distribution** is another special case of the gamma distribution, and is parameterized by a *shape parameter* ${tex`k \in \mathbb{Z}_+`} and a *rate parameter* ${tex`\lambda > 0`}. If we have ${tex`\{X_1, X_2,\, \cdots, X_k\} \stackrel{\mathrm{\tiny{i.i.d.}}}{\sim} \mathrm{Exponential}(\lambda)`}, then ${tex`\sum_{i=1}^k X_i \sim \mathrm{Erlang}(k, \lambda)`}. The Erlang distribution is used to model the time until multiple "memoryless" events occur.

*analytical* mean interarrival: ${tex`~~~~~~~~~~~\,k\lambda = ${ Math.round(param[0] * param[1] * 100) / 100 }`} *empirical* mean interarrival: ${tex`~~~~~~~~~~~~~~~~\,\,\,\approx`} ${tex`${ Math.round(diagram[1] * 100) / 100 }`}

*analytical* mean spread: ${tex`~~~~~~~~\,\,(k+1)\lambda = ${ Math.round((param[0] + 1) * param[1] * 100) / 100 }`} *empirical* mean spread: ${tex`~\,~~~~~~~~~~~~~~~~~~~~~~~~\approx`} ${tex`${ Math.round(diagram[2] * 100) / 100 }`}

`; break; default: return md`you should not be reading this!`; } }