1. Probability — Venn
$P(A\cup B) = P(A) + P(B) - P(A\cap B)$. Independence holds when $P(A\cap B)=P(A)P(B)$.
2. Discrete distributions
PMF + CDF. Mean, variance, mode shown — try to predict them before you read.
3. Continuous distributions
PDF in colour, CDF overlay (toggle). Integrate the PDF over an interval to read a probability.
4. Sum of independent variables
Density of $X+Y$ is $(f_X * f_Y)$, the convolution. Two uniforms → triangle. Two normals → wider normal.
5. Central Limit Theorem
Sample-mean histogram from any reasonable source converges to $\mathcal{N}(\mu, \sigma^2/n)$.
6. Bayes — base-rate fallacy
1000 people. Drag prior, sensitivity, specificity. Watch how rare diseases stay rare even after a positive test.
7. Maximum likelihood
Fix some data. Slide the parameter. The likelihood peaks at the MLE — for an i.i.d. normal sample, that's the sample mean.
8. Confidence intervals
100 random samples from $\mathcal{N}(0,1)$, 100 intervals. Long-run coverage should match nominal.
9. Hypothesis test & p-value
Red shade: Type I (α). Orange shade: Type II (β). Drag $\delta$ and $c$ to feel the trade-off.
10. t-distribution vs Normal
Heavier tails when $df$ is small. As $df\to\infty$ it converges to $\mathcal{N}(0,1)$.
11. Linear regression
Drag points. The OLS line minimises $\sum(y-\hat y)^2$. Watch $R^2$ collapse when you add an outlier.
12. Monte Carlo $\pi$
Fraction of darts inside the quarter circle, times 4. Error shrinks like $1/\sqrt N$.
13. Random walks
Symmetric walks spread like $\sqrt t$; a tiny drift dominates eventually.