Statistics, Probability and Discrete Mathematics — BMATB301 (Modules 1–5): quick revision, 30 bites

The 5-Module Map of BMATB301

Five modules, one exam strategy.

The paper is split into **5 modules**; you answer **any FIVE full questions**, with **at least ONE from each module**. Knowing the map is half the battle.

Takeaway. Pick your strongest question from each module first, then fill the fifth from your best module.

Normal Equations for Curve Fitting

Fit any polynomial with these equations.

For $y = a_0x^2 + a_1x + a_2$, minimize the sum of squared errors. The normal equations are:

Takeaway. Build a table of $\sum x, \sum x^2, \sum x^3, \sum x^4, \sum y, \sum xy, \sum x^2y$ first — then solve the 3×3 system.

Correlation Coefficient from Regression Lines

Two regression lines hide the correlation.

Given regression lines $b_{yx}$ and $b_{xy}$, the correlation coefficient is the **geometric mean** of the two slopes.

Takeaway. $r = \sqrt{b_{yx} \cdot b_{xy}}$ — and the means $(\bar{x}, \bar{y})$ lie on BOTH regression lines.

Fitting $y = ax^b$ via Logarithms

Log it, linearize it, fit it.

Takeaway. Any power law becomes linear after taking logs — that's the whole trick.

Spearman's Rank Correlation

Ranks disagree? Measure it.

Takeaway. $\rho$ ranges from $-1$ to $+1$; $\rho = 1$ means perfect agreement between judges.

Poisson Distribution — Rare Events

Rare events, big n, tiny p.

Use Poisson when $n$ is large and $p$ is small, with $\lambda = np$.

Takeaway. For 'more than 2', always use the complement: $1 - [P(0)+P(1)+P(2)]$.

Exponential Distribution for Waiting Times

Mean 10 ms? Then λ = 0.1.

Takeaway. Mean = $1/\lambda$. Convert mean to $\lambda$ BEFORE plugging into any formula.

Binomial Distribution — Fixed Trials

n trials, p success, done.

Takeaway. Binomial = fixed $n$, independent trials, constant $p$ — otherwise use Poisson.

Normal Distribution — Mean & SD from Percentages

Two percentages reveal μ and σ.

Convert given percentages to z-scores using the standard normal table, then solve two simultaneous equations.

Takeaway. Two tail conditions → two equations → unique $(\mu, \sigma)$.

Chi-Square Goodness of Fit

Observed vs expected — one number tells all.

Degrees of freedom = (number of categories) − 1 (or − 1 more if parameters are estimated).

Takeaway. Small $\chi^2$ → data fits theory; large $\chi^2$ → reject the hypothesis.

z-test vs t-test — Which One?

Big sample or small? Choose wisely.

Takeaway. If $\sigma$ is unknown and $n$ is small, always use the t-test with $n-1$ degrees of freedom.

Sampling Distribution of the Mean

Sample mean has its own distribution.

Takeaway. Standard error shrinks as $\sqrt{n}$ grows — bigger samples give tighter estimates.

Testing a Hypothesis — The 5-Step Ritual

Five steps, every hypothesis test.

Takeaway. If |computed| > |critical|, reject $H_0$. Otherwise, fail to reject.

Coin Fairness — How Many Heads?

Fair coin? Count the heads.

For 64 tosses, $\mu = 32$, $\sigma = \sqrt{64 \cdot 0.5 \cdot 0.5} = 4$. Reject fairness if $|z| > 1.96$.

Takeaway. Fairness holds when $X$ lies within $\mu \pm 1.96\sigma$, i.e., roughly 24 to 40 heads.

Validity of an Argument in Logic

Premises true → conclusion must follow.

An argument is **valid** if the conclusion is true whenever ALL premises are true. Validity is about structure, not about real-world truth.

Takeaway. Valid ≠ true. A valid argument can have false premises — it just can't have true premises with a false conclusion.

Tautology Check via Truth Table

All T's in the last column? Tautology.

A **tautology** is a compound proposition that is true for every assignment of truth values to its variables.

Takeaway. Tautologies are the logical backbone of valid inference rules.

Direct vs Indirect Proof

Two ways to prove the same thing.

Takeaway. Indirect proof proves $P \to Q$ by showing $\neg Q \to \neg P$ (contrapositive).

Partial Order — Reflexive, Antisymmetric, Transitive

Three properties make a poset.

Takeaway. A set with a partial order is called a **poset** $(A, R)$.

Drawing a Hasse Diagram

Strip the clutter, keep the order.

Takeaway. A Hasse diagram is the minimal graph that still captures the entire partial order.

Equivalence Relation — The Three Tests

Reflexive, symmetric, transitive.

Takeaway. Equivalence relations partition a set into disjoint equivalence classes.

Inverse Function Test

Compose both ways — get identity?

Takeaway. One-sided composition is not enough — check BOTH directions.

The #1 Regression Sign Trap

Wrong sign, wrong answer.

Takeaway. Sign of $r$ = sign of the regression coefficients. Never lose it.

Poisson λ Trap

Mean or λ? Don't mix them.

Takeaway. Exponential: mean = $1/\lambda$. Poisson: mean = $\lambda$. Different conventions!

Exam Strategy — Answer Any FIVE

Five questions, one from each module.

Takeaway. Strategic question selection beats last-minute cramming.

Remember the Chi-Square Formula

O minus E, squared, over E.

Chant it: **"O minus E, squared, over E — sum it up, that's $\chi^2$."**

Takeaway. If $\chi^2_{calc} < \chi^2_{table}$, the fit is good — accept the hypothesis.

Why Correlation ≠ Causation

Two numbers move together — so what?

A high $|r|$ only means two variables move together **linearly**. It does NOT prove one causes the other — a hidden third variable could drive both.

Takeaway. $r$ measures linear association, not cause and effect.

Where These Distributions Show Up

Real systems, real distributions.

Takeaway. Recognizing the real-world pattern tells you which distribution to use.

Complement Trick for 'At Least' Problems

At least one? Flip it.

Takeaway. Whenever you see 'at least', reach for the complement.

Degrees of Freedom — Don't Guess

Wrong df, wrong critical value.

Takeaway. Always write down $df$ explicitly before looking up the critical value.

Matrix Representation of a Relation

Relations become 0-1 matrices.

For a relation $R$ on a finite set $A = \{a_1, \ldots, a_n\}$, the matrix $M(R)$ has entry $m_{ij} = 1$ if $(a_i, a_j) \in R$, else $0$.

Takeaway. The digraph and the matrix are two views of the same relation.