Ugg throw bones, learn what tomorrow bring

Ugg sit by fire. Ugg want know: turmeric milk fix sore throat, or just luck dressed like finding? To know, Ugg must learn probability — the count of how often a thing happen.

Two tribe fight over "how likely"

Probability is Ugg guess for how often thing happen. Two tribe argue what the word mean.

Frequentist probability — probability is the long-run frequency. Flip shiny stone a billion time, half land head, so P(head) = 0.5. Good: same answer for every caveman, no opinion. Bad: weird for one-time thing. "70% rain today" must mean "of all days like this one, it rain on 70%." Try say that to friend holding umbrella.

Bayesian probability — probability is Ugg degree of belief, change when new footprint appear. "P(Carlsen win) = 0.7" mean Ugg feel 70% sure from what Ugg already know. Good: work for one-time thing. Bad: lean on Ugg starting hunch. This course mostly use frequentist.

Probability go one way, statistics go the other

Probability flow model → data: Ugg know stone is fair, so Ugg compute P(two tail in row) = 0.25.

Statistics flow data → model: Ugg see hunter win 3 fight of 3, so Ugg ask — is the true win-chance really 0.5, or bigger? Most of course go this second way. Old chess story warn Ugg: every method make assumption. Fair-coin guess said Fischer 50%; he win 6–0. Wrong assumption give confident-but-wrong answer. Choose model with humble heart.

Words for the hunt

Independent events — knowing one tell Ugg nothing about the other. $P(A\cap B)=P(A)\cdot P(B)$. Stone-flips independent. Rain and umbrella, NOT.

i.i.d. (independent and identically distributed) — all draws independent AND all from same rock-pile. This the ground every test in course stand on. Same hunter tested twice? Not i.i.d. — need special method (paired test).

Sample vs population — population is all mammoth in the world; sample is the few Ugg actually count. Ugg never see all; Ugg guess the whole from the few.

Sampling distribution — this the secret heart. Grab sample, compute a number (the mean). Grab another sample, compute again. Do many time. The spread of THOSE numbers is the sampling distribution. Every test compare Ugg number to this spread.

Shape-of-chance rocks

PMF (probability mass function) — for countable thing, gives $P(X=k)$ for each value.

PDF (probability density function) — for smooth thing, $f(x)$. Chance of an EXACT value is zero. Only a range hold chance: $P(a\le X\le b)={\int }_a^{b}f(x)dx$. Area under whole PDF = 1.

CDF (cumulative distribution function) — $F(x)=P(X\le x)$, climb from 0 up to 1.

Ugg warn: PDF can stand taller than 1 — it is density, not chance. PMF never go above 1.

The tribe of distributions

Bernoulli(p) — one yes/no throw. P(1)=p, P(0)=1−p.

Binomial(n,p) — add $n$ i.i.d. Bernoulli. $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$. Mean $np$, variance $np(1-p)$. Ugg warn: variance is $np(1-p)$, NOT $np(1-p{)}^2$. In R: dbinom(6,10,0.7) ≈ 0.200.

**Normal $\mathcal{N}(\mu ,{\sigma }^2)$** — the bell hill, two knobs: mean $\mu$, spread $\sigma$. $f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$. Empirical rule: ~68% inside $\pm 1\sigma$, ~95% inside $\pm 2\sigma$, ~99.7% inside $\pm 3\sigma$ — that why beast beyond 2–3σ is called outlier. Make any Normal standard with the z-score $Z=(X-\mu )/\sigma$: slide to mean 0, squeeze to SD 1.

t-distribution — like Normal but fatter tail, one knob (degrees of freedom). Use when sample small AND true $\sigma$ unknown — estimating it add uncertainty, the fat tail. As df grow big, t become Normal. Ugg warn: small sample, unknown σ — use t, not Normal, else Ugg reject too often.

**Chi-square ${\chi }_k^2$** — square $k$ standard Normals and add: $Z_1^2+\dots +Z_k^2$. Always ≥ 0, skew right. Mean $k$, variance $2k$.

F-distribution — ratio of two scaled chi-squares: $F=\frac{U/d_1}{V/d_2}$. Skew right, ≥ 0, two df. Lives in ANOVA. One tail only.

R four magic letters

For any distribution xxx: d density/PMF, p cumulative CDF $P(X\le q)$, q quantile (inverse CDF), r random draw. So qnorm(0.975) ≈ 1.96, and pbinom(4,10,0.7) ≈ 0.047.

The great spell — Central Limit Theorem

Central Limit Theorem (CLT) — the biggest magic in whole course. Given big-enough sample, the sampling distribution of the mean become Normal — no matter what shape the population, so long as it have finite variance.

Population can be lumpy, skewed, ugly. The mean-of-a-sample still make a bell. THIS why parametric test work so wide — they need the sampling distribution Normal, not the raw data. Ugg warn loud: CLT is about the spread of the mean, NOT the raw data turning Normal.

Two gift come with it. One: the sample mean $\bar{X}$ is unbiased, $\mathbb{E}[\bar{X}]=\mu$. Two: the spread of that mean is the SEM (Standard Error of the Mean):

Bigger $n$ → smaller SEM → sharper spear. The famous $\sqrt{n}$: to halve SEM, Ugg need 4× the data. Ugg warn: it is $\sigma /\sqrt{n}$, NOT $\sigma /n$.

Cousin rule, Law of Large Numbers: as $n$ grow, $\bar{X}\to \mu$. LLN say Ugg arrive at truth; CLT say the path there is bell-shaped. When sample tiny AND data very skewed, CLT not kick in yet — use non-parametric test. Rule of thumb: $n>25$–$30$ per group enough for mild skew.

Ugg remember

• CLT rule all: sampling distribution of the mean → Normal as $n$ grow, for any population with finite variance. It about the mean, NOT the raw data.
• **SEM = $\sigma /\sqrt{n}$**. To halve it, need 4× the data. Never write $\sigma /n$.
• i.i.d. hold up every test. Independent mean $P(A\cap B)=P(A)P(B)$; same beast measured twice break it.
• Know the tribe: Bernoulli (one throw), Binomial (mean $np$, var $np(1-p)$), Normal (68/95/99.7), t (fat tail), ${\chi }^2$ (mean $k$, var $2k$), F (ratio of chi-squares).
• Continuous exact value has chance 0 — ask for a range. PDF density can top 1; PMF cannot. R letters: d/p/q/r.