# Administrative Information - **Class Meeting**: Friday, October 16, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 20) - Correlation, and two limits]] - 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]] > [!info]- Admin. Notes > Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes. > > - 🎯 **[LR] [Day 20 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121790)** — 4 questions drawn from Wednesday's material, unlimited attempts, highest kept — due **Friday 10/16, 11:59pm** > - 🎯 **[LR] [Day 20 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121791)** — one free response, no wrong answers — due **Friday 10/16, 11:59pm** > - 🎯 **[LR] [Day 21 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121823)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Monday 10/19, 11:59pm** > - 🎯 **[LR] [Day 21 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121824)** — one free response, no wrong answers — due **Monday 10/19, 11:59pm** > - ✍️ **[Workbook check-in, digital copy](https://elearning.mines.edu/courses/81632/assignments/590067)** — upload your returned sheet to Canvas — due **today, Friday 10/16, 11:59pm** > - 📬 **MP2 / MA1 feedback went out today** as Canvas messages (not tied to the assignments). Every message ends with the same two notes: *how I read a paper*, and *if office time is needed, email or come by*. A couple of MAs are still missing; the feedback stands either way. > - 🕒 **Office block for 310:** Fridays at 3pm are now held for this class (the by-appointment slot is locked out for anyone else); Tuesday/Thursday blocks will be announced as they open. Other times — just email. > - 🎯 **[MR] [Media Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121701)** · **[Statement](https://elearning.mines.edu/courses/81632/quizzes/121702)** · [[The Breakdown of Linear Stability Analysis and Critical Slowing Down|the page]] — due **Friday 10/23, 11:59pm** > - 📝 **[MP3 — Initial Model and Test](https://elearning.mines.edu/courses/81632/assignments/590872)** and **[MA2 — Source and Background](https://elearning.mines.edu/courses/81632/assignments/590873)** — both due **Monday 10/26, 11:59pm** ([[MATH310F26 - Project Work|Project Work]]; moved from Friday 10/23; > - 🎯 **[WB] [Workbook 11](https://elearning.mines.edu/courses/81632/assignments/588368)** · [[MATH310F26_WB11.pdf|blank]] · [[MATH310F26(Workbook 11 Companion) - Three outcomes, two moments|companion]] — due **Friday 10/30, 11:59pm** > - 🎬 [[MATH310F26 - Random numbers, two videos]] — the guided videos for this unit, if you haven't yet --- # Deliverables - 🎯 **[LR] [Day 21 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121823)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Monday 10/19, 11:59pm** - 🎯 **[LR] [Day 21 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121824)** — one free response, no wrong answers — due **Monday 10/19, 11:59pm** - ✍️ **[Workbook check-in, digital copy](https://elearning.mines.edu/courses/81632/assignments/590067)** — due **tonight, Friday 10/16, 11:59pm** - 📝 **[MP3 — Initial Model and Test](https://elearning.mines.edu/courses/81632/assignments/590872)** and **[MA2 — Source and Background](https://elearning.mines.edu/courses/81632/assignments/590873)** — both due **Monday 10/26, 11:59pm** (moved from Friday 10/23) Questions and statement are due Monday 10/19 at 11:59 pm. --- # Lecture Boards + Transcript + GenAI **Twenty-first meeting, October 16 — [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution) and heavy and fat tails, continuum limits.** Board header: *21 | 10/16/26: Cauchy Distro. and heavy and fat tails, Continuum limits.* "Half of the clocks are saying one," so a short week ends on a Friday that arrived all of a sudden. Three movements promised up front, and two and a half delivered: finish the Cauchy distribution — the [expected value](https://en.wikipedia.org/wiki/Expected_value) that isn't there — then the nomenclature of heavy versus fat tails, and then the start of a **continuum limit**: take the [Plinko](https://en.wikipedia.org/wiki/Plinko) board, with its rows of pegs as time and the gaps between pegs as space, smoosh both until they are tiny, and ask what you get. That is the road back to *why* the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem) sits underneath additive random processes at all. ("There are a variety of ways to do it — you can perform a limit on a [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution) and make stuff happen. Fun derivation, but it doesn't really give us any insight into the underlying process." The one chosen fits this audience: a 307 in everyone's past and a [PDE](https://en.wikipedia.org/wiki/Partial_differential_equation) class in many futures.) It ends with a [difference quotient](https://en.wikipedia.org/wiki/Difference_quotient) on the board and a promise of the [heat equation](https://en.wikipedia.org/wiki/Heat_equation) on Monday. ## Housekeeping About five minutes before class, feedback on **MP2 and MA1** went out as Canvas messages — read holistically, so sent as messages rather than attached to the assignments; a couple of MAs are still missing, but the feedback stands for where to take the next step. Every message carries the same two things after the signature. One: **how I read a paper** — "I realized I was talking about this frequently, so I just joined it together and put it at the end of every one"; many groups are reading academic literature now, and this is how Scott was taught to read it ("if you have something different, that's totally fine too"). Two: the Canvas pipeline is reaching the point where it is "not going to be as useful or efficient going forward, because by and large things are becoming very complicated on your side" — so **if office time is needed, email or come by**. A refinement negotiated live: dedicated blocks where 310 has the office and no one else does. Not Mondays ("Mondays are their own special thing"); Fridays, yes — "end the week off on some math" — and **Fridays at 3pm** had enough hands that the by-appointment slot is now locked out for this class. Tuesdays and Thursdays are the other open territory for the MWF-locked math group, so those blocks will appear as the schedule allows. Other times: email; it's a small class, and accommodations are easy "if I can leave my doggie for that time." ## Recall: moments, and the language for them **Not all probability distributions have well-defined first and second moments.** The language has been loose, so a pass at fixing it. The **first moment** is $\mathbb{E}[X]$ — what statisticians would call the expected value or expectation. Continuing the pattern, the **second moment** is $\mathbb{E}[X^{2}]$, and *that is not the [variance](https://en.wikipedia.org/wiki/Variance)*. But given $\mathbb{E}[X]$ and $\mathbb{E}[X^{2}]$, is there enough to make the variance? "Yeah, 100%": $\mathrm{Var}[X]=\mathbb{E}[X^{2}]-\big(\mathbb{E}[X]\big)^{2},\qquad \mathrm{Var}[X]=\mathbb{E}\big[(X-\mu_{X})^{2}\big].$ So $\mathbb{E}[X^{2}]$ is the **raw second moment** and $\mathbb{E}[(X-\mu_{X})^{2}]$ is the **centered second moment** — the second moment taken about the first. The reason for preferring *moments* over *mean and variance* is portability: the first moment is the **[center of mass](https://en.wikipedia.org/wiki/Center_of_mass)** — the point on the marker where a finger holds it up, the dot that traces the parabola when the marker is thrown — and the centered second moment is the **[moment of inertia](https://en.wikipedia.org/wiki/Moment_of_inertia)**, the spread about that center. The [ice skater](https://en.wikipedia.org/wiki/Angular_momentum): arms out, spinning; pull the arms in and "is it easier to spin me or harder?" Easier, and the skater speeds up. Statistics calls these central tendency and spread; the formulas are the same ones that show up everywhere else. Why it matters here: the central limit theorem says that additive [stochastic processes](https://en.wikipedia.org/wiki/Stochastic_process) have, at least asymptotically, a describable ensemble distribution — *provided* the steps have well-defined first and second moments. Lose those and "there's no expectation of seeing something like the central limit theorem or [normal distributions](https://en.wikipedia.org/wiki/Normal_distribution) come out." **The CLT would not apply to a random walk built by such a random variable.** And where was the process left on Wednesday? At the note that something concrete would show it. ## The expected value that isn't there Let $X\sim\mathrm{Cauchy}(x_{0}=0,\gamma=1)$: hump centered at the origin, [scale parameter](https://en.wikipedia.org/wiki/Scale_parameter) set to one, "because really what I want to see is the moment process fail." The expectation is $\mathbb{E}[X]=\int_{-\infty}^{\infty}x\,\rho(x;0,1)\,dx,$ the $x$ out front weighting the density by position. (An aside, written and erased: the second moment would carry $x^{2}$ instead, which *shrinks* the contribution of anything between $0$ and $1$ and *inflates* anything beyond — which is exactly why it measures spread, emphasizing what is far from the center over what is near it.) Three limits hide in that integral — Calc 1 [improper integrals](https://en.wikipedia.org/wiki/Improper_integral), both bounds and the continuous sum itself — so make the bounds explicit: let $a\to-\infty$ and $b\to+\infty$ as two separate processes, and integrate from $a$ to $b$. With the Day 20 density, $\mathbb{E}[X]=\lim_{\substack{a\to-\infty\\ b\to\infty}}\int_{a}^{b}x\cdot\frac{1}{\pi(1+x^{2})}\,dx.$ "I'm going to go mano a mano here and not look at my notes." How to do the integral? From the room: [$u$-substitution](https://en.wikipedia.org/wiki/Integration_by_substitution) on the denominator. $u=1+x^{2}$, $du=2x\,dx$, so $x\,dx=\tfrac12\,du$; the $1/\pi$ comes out front, the $\tfrac12$ joins it, and the bounds have to move into the $u$-variable — a premature set of $x$-bounds got corrected on the board — $x=a$ gives $u=1+a^{2}$, $x=b$ gives $u=1+b^{2}$: $=\lim_{\substack{a\to-\infty\\ b\to\infty}}\frac{1}{2\pi}\int_{1+a^{2}}^{1+b^{2}}\frac{du}{u}=\lim_{\substack{a\to-\infty\\ b\to\infty}}\frac{1}{2\pi}\Big[\ln|u|\Big]_{1+a^{2}}^{1+b^{2}}=\lim_{\substack{a\to-\infty\\ b\to\infty}}\frac{1}{2\pi}\ln\!\left(\frac{1+b^{2}}{1+a^{2}}\right).$ (The notes-consult moment: "Yo, Scott, why are you skipping so many steps in your notes? That means you have to think about stuff." The absolute values are "pretty chill" — positive plus positive at both ends — and the difference of logs became the log of the quotient, from the room.) No need to go back to $x$: "who cares about that? I'm ready to go from right here." **Now the move nobody likes.** $a$ says how far to go to the left of the origin, $b$ how far to the right, and they were kept separate because those are two processes that are not required to be coupled. "This introduction of this next step will be arbitrary. But it's supposed to show that this thing is really broken. So as long as I show that it's broken for one case, I don't need to think about a lot of other cases." Let $a=-k\,b,\qquad k>0.$ As $b\to\infty$, $a\to-\infty$ too — just at some multiple of $b s rate; $k=1$ is the symmetric march out, and any other $k$ is an asymmetric one. Substitute, and there is only one limit left: $\frac{1}{2\pi}\lim_{b\to\infty}\ln\!\left(\frac{1+b^{2}}{1+k^{2}b^{2}}\right)=\frac{1}{2\pi}\ln\!\left(\frac{b^{2}}{k^{2}b^{2}}\right)=\frac{1}{2\pi}\ln\!\left(\frac{1}{k^{2}}\right)=-\frac{\ln k}{\pi}.\qquad(\ast)$ ($b$ getting real big makes $b^{2}$ real big; "is adding 1 to that going to matter? Not a bit" — and $k^{2}$ is a fixed number, so the $+1$ next to $k^{2}b^{2}$ doesn't matter either; the $b^{2} s cancel.) The limiting result depends on $k$. "If I choose $k=2$ and you choose $k=3$, we're going to get a different limit. And in fact, for any $k$ that's chosen, we're going to get a different limit." A quantity that can be steered to any value by the choice of how to scope left and right has no unique value to settle on — "and so we ought to not define it." ($a$ and $b$ are meant to reach *the* infinity of the real line, the [uncountable](https://en.wikipedia.org/wiki/Uncountable_set) one; the formula can't tell which road was taken there, and the answer depends on the road.) > [!abstract] $(\ast)$ There is no single well-defined limit $\;\Longrightarrow\;\mathbb{E}[X]$ DNE. > It depends entirely on how you scope to the right and to the left. And once the first moment goes, the chain of command goes with it: the variance is $\mathbb{E}[X^{2}]-(\mathbb{E}[X])^{2}$, the second piece has no definition, so the left-hand side has none either. "The second I lose one of these moments, the rest of the moments are going to be lost, too." ## Heavy, fat, and a black swan While PowerPoint loaded ("always strangely time-consuming"), the picture from Day 17 came back: blue the normal distribution, orange the Cauchy. Both have humps. For the normal, the hump sits at the mean; for the Cauchy, "I can't call that the mean, because I lost my handle on the calculation which defines the mean." The difference is in the decay. Blue is governed by $e^{-x^{2}}$ — it comes off its high and decays away a ton, real fast, so that by $4$ it is nearly gone; that is the [68–95–99.7 rule](https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule), with the area under the curve wrapping itself up quickly as you move out. Orange does not do this: there is a lot of accumulated probability still to be had far out, and it decays as a **[power law](https://en.wikipedia.org/wiki/Power_law)**. The nomenclature, then: a distribution has **[heavy tails](https://en.wikipedia.org/wiki/Heavy-tailed_distribution)** if its tails decay *slower than an [exponential](https://en.wikipedia.org/wiki/Exponential_decay)*; it has **[fat tails](https://en.wikipedia.org/wiki/Fat-tailed_distribution)** if they decay *like a power law*. (Fat is the stronger condition; the Cauchy is both.) The slide that shows the consequence: compute the "mean" of each density numerically from $-a$ to $2a$ — note the asymmetric scoping, twice as far to the right as to the left — and plot it against $a$. **Blue:** small $a$ grabs the hump lopsidedly and the estimate rises to a peak near $a\approx 1$; then, as more of the curve comes in, the number *decreases* and converges to $0$, "which is what you would expect for the blue humpy thing," even with the window twice as wide on the right. **Orange:** as $a$ grows, the Cauchy keeps handing back a nonzero mean that keeps rising; the more you integrate to the right, the more tail you collect, and the numerics drift the mean out to the side. "That's why it's not defined — it won't localize to a spot according to our formula." > [!example]- The slide, re-run here > ![[day21_asymmetric_means.png]] > Left: the two densities, with one $[-a,2a]$ window shaded. Right: $\int_{-a}^{2a}x\,\rho(x)\,dx$ against $a$. The normal's estimate peaks at $0.188$ near $a=0.95$ and dies to $0$; the Cauchy's rises and levels off at $\ln 2/\pi\approx 0.2206$ — which is exactly $(\ast)$ with $k=\tfrac12$. The "mean" is set by the scoping, not by the distribution. **Why would anyone want heavy tails?** Because they correspond to **rare events**, and rare events definitely happen. In finance they are **[black swan](https://en.wikipedia.org/wiki/Black_swan_theory)** events — swans are normally white, and a [genetic mutation](https://en.wikipedia.org/wiki/Mutation) every now and then produces a black one. The textbook black swan: human height and weight are, generally speaking, normally distributed, and then there is **André René Roussimoff — [Andre the Giant](https://en.wikipedia.org/wiki/Andr%C3%A9_the_Giant)** — at 7′4″ and 520 lb. On a normal model, the slide's probability of sampling a height within $\pm 10\%$ of his is $10^{-248}$: "that number doesn't even make sense — it's not 0, but it's like 0." The atoms in the [observable universe](https://en.wikipedia.org/wiki/Observable_universe) are something like $10^{80}$; nothing comparable exists. The weight is kinder: $10^{-9}$, in the company of hitting the [Powerball](https://en.wikipedia.org/wiki/Powerball) twice in a lifetime, being [struck by lightning](https://en.wikipedia.org/wiki/Lightning_strike) several times in one year, or guessing a 30-digit number. Yet Andre existed, and the stock market did lose about a quarter of its value on a [Black Monday](https://en.wikipedia.org/wiki/Black_Monday_(1987)) in the eighties. These things occur; the normal distribution does not prescribe them; a Cauchy-like tail decays slowly enough to allow them. When do you break out a heavy-tailed model instead of the normal that mostly works? "I don't really know the answer to that. I'm not sure if there is one." A replay of the Cauchy walk from Day 17: the lump is the lump, and then some outlying walker decides to jump ten thousand steps away — walk, walk, walk around the bump, then "boom, 1,500 units all of a sudden," settle there, then another five to seven hundred. "Central limit theorem isn't binding you with the chain or anything." > [!example]- The Cauchy walkers, from Day 17 > ![[day17_cauchy_walkers.png]] > Fifty of 5000 walkers with unit-scale Cauchy steps; most hug the center, and single steps of thousands of units arrive without warning. (Re-run on the Day 17 page.) **Andrew's question** — in place of mean and variance, are there proxies? Yes: the parameters generally fit the intuition, they just can't be called by the old names. $x_{0}$ is where the hump is placed — the [*location*](https://en.wikipedia.org/wiki/Location_parameter) (placement) parameter, which would be the mean for a normal; $\gamma$ controls how wide or spread out the distribution is — the *scale* parameter, which can't be called a variance because the variance isn't defined. "The names don't apply if you were to lose control in that way." ## The continuum limit, started (Chalk again — "a delightful sound," next to the memory of the squeaky boards of the first week; also, the Broncos–Seahawks game nobody told the instructor about.) To conclude the random-walk discussion: **why should the central limit theorem come up at all?** It helped think about what should naturally happen on the Plinko board, and the derivation that fits this room best is the one that runs through a PDE. Consider an organism that must obey the following. Build a **[sample space](https://en.wikipedia.org/wiki/Sample_space)** $\Omega$, as when the Plinko talk started, with three things in it: **move left, move right, or do nothing** — with left and right by the same standard amount, to keep it simple — and suppose the organism picks from $\Omega$ with certain probabilities. Now ask for the **proportion of walkers at location $x$ at time $t+\Delta t$**. Time updates were count variables before; now the old time plus one count is $t+\Delta t$, "more along the lines of the calculus that we've been exposed to." The diagram: a horizontal line for time $t+\Delta t$ with the location $x$ marked on it, and a horizontal line for time $t$ below it, $\Delta t$ earlier. Anyone standing at $x$ at the later time must have made their move from the earlier line, and the sample space says where from — **three sources**: the do-nothing crowd straight up from $x$ (green); the group at $x-\Delta x$, if $\Delta x$ is the step width, who chose to go right (purple); and the group at $x+\Delta x$ who chose to go left (red). Let $u$ be the normalized count of walkers, and write what arrives: $u(x,t+\Delta t)=P_{r}\,u(x-\Delta x,t)+P_{0}\,u(x,t)+P_{l}\,u(x+\Delta x,t).$ "Leave yourself some space in front of these $u s" — because the sum of everything on the earlier line is *not* what arrives: the walkers at $x-\Delta x$ had three options, and only the fraction $P_{r}$ that went right made it. So the formula is cooked with $P_{r}$ for the walkers on the left who went right, $P_{0}$ for those in the middle who stayed, and $P_{l}$ for those on the right who went left. One thing not yet written down but concluded from the covered-up sample space: three outcomes, each with a probability, and if the walker must do *something* at every moment, those probabilities **sum to one**. Replace the green term: $u(x,t+\Delta t)=P_{r}\,u(x-\Delta x,t)+P_{l}\,u(x+\Delta x,t)+\big(1-P_{r}-P_{l}\big)\,u(x,t).$ And then the glimpse, with the clock gone. "Forget about all this purple stuff. Look at this guy." There is a $u(x,t)$ on the right. Send it to the left: $u(x,t+\Delta t)-u(x,t)$ — the change in $u$ over the time step. Divide by $\Delta t$ and let $\Delta t$ get real small — "do you see a [derivative](https://en.wikipedia.org/wiki/Derivative) coming out of that?" That is the continuum limit: the Plinko peg widths $\Delta x$ and the row spacing $\Delta t$ shrink together. **Next time: derive those [finite differences](https://en.wikipedia.org/wiki/Finite_difference), shrink them until they become derivatives, and find the heat equation.** > [!success]- Numbers checked in one sandbox run > Symbolically, $\int_{a}^{b}\frac{x\,dx}{\pi(1+x^{2})}=\frac{1}{2\pi}\ln\frac{1+b^{2}}{1+a^{2}}$; with $a=-kb$ the limit as $b\to\infty$ is $-\ln(k)/\pi$, i.e. $\frac{1}{2\pi}\ln(1/k^{2})$ as on the board: $0$ for $k=1$, $+0.2206$ for $k=\tfrac12$, $-0.2206$ for $k=2$ — a different number for every $k$. The symmetric limit $\int_{-R}^{R}x\rho\,dx$ is identically $0$ ([odd integrand](https://en.wikipedia.org/wiki/Even_and_odd_functions)), which is why the *symmetric* march alone would have hidden the problem. The density itself integrates to $1$, and the raw second moment $\int_{-R}^{R}x^{2}\rho\,dx=\frac{2}{\pi}(R-\arctan R)$ grows without bound — so $\mathrm{Var}[X]$ fails even without the $\mathbb{E}[X]$ chain. The slide's asymmetric-window curves: for the normal, $\int_{-a}^{2a}x\varphi\,dx=\varphi(a)-\varphi(2a)$, peak $0.188$ at $a=0.95$ (the slide's "0.25-ish, a little lower"), then $\to 0$; for the Cauchy, $\frac{1}{2\pi}\ln\frac{1+4a^{2}}{1+a^{2}}$, which is $0.146$ at $a=1$, $0.219$ at $a=10$, and converges to $\ln 2/\pi=0.2206$ — it rises and *levels off at a value set by the 2:1 scoping*, rather than growing forever; quadrature agrees to all printed digits. The normal's 68–95–99.7: $0.6827$, $0.9545$, $0.9973$ (the "68.7" said aloud was a slip of the tongue). The Andre numbers ($10^{-248}$, $10^{-9}$) are the slide's; a rough normal model of adult male height (mean $69$ in, s.d. $2.9$ in) puts $88$ in at $z\approx 6.5$, and $520$ lb at $z\approx 7$ on a mean-$199$, s.d.-$45$ model, so the $10^{-9}$ for weight is the right order and the $10^{-248}$ for height depends on a far narrower model than the one here — either way, nothing a normal would ever hand you. > [!example]- Board scans — October 16, 2026 > ![[MATH310F26-Day21-Board-1.jpg]] > ![[MATH310F26-Day21-Board-2.jpg]] > ![[MATH310F26-Day21-Board-3.jpg]] > ![[MATH310F26-Day21-Board-4.jpg]] > ![[MATH310F26-Day21-Board-5.jpg]] > > 📄 [[MATH310F26-Day21-Boards-2026-10-16.pdf|Full board capture (5 pages, PDF)]]