Lecture 4 Continuous Random Variables & Normal Distribution – Comprehensive Lecture Notes

Course Orientation

• MATH1041 Statistics for Life and Social Sciences – Chapter 4 notes (Term 2 2025)
  • Overarching aim: introduce statistics as the science of collecting, analysing and interpreting data.
  • Chapter 4 focus: Continuous Random Variables & the Normal Distribution.
  • Four linked lectures:
  • L1 Continuous random variables & density curves.
  • L2 Normal distributions.
  • L3 Probabilities & quantiles from a normal distribution.
  • L4 Simulation examples.
  • Textbook alignment: Moore et al. (2021) Section 4.3 & Section 1.4.

Quick Revision of Discrete Random Variables (context from previous chapter)

• Discrete rv’s: take countable values (finite or countably infinite list).
• Binomial model recap (independent Bernoulli trials):
  • $P(X=x)=\binom{n}{x}p^{x}(1-p)^{n-x},\;x=0,1,\dots,n$
  • Mean $\mu<em>X=np$, variance $\sigma</em>X^{2}=np(1-p)$.
• Rules for linear combinations (any r.v.’s):
  • $E(a+bX)=a+bE(X)\,,\qquad Var(a+bX)=b^{2}Var(X)$.
  • If independent, $Var(X\pm Y)=Var(X)+Var(Y)$ (NOT true for SD’s nor dependent RV’s).
• Clarification of “mean” terminology:
  • True (population) mean $\mu<em>X$; sample mean $\bar X$ (a rv); observed mean $\bar x</em>n$ (a number).

Lecture 1 – Continuous Random Variables & Density Curves

Definitions & Conceptual Bases

• Uncountable set: cannot be put in a list (e.g. all reals in [0,1]).
• Continuous rv: can take every value in an interval ⇒ range uncountable.
• Practical criterion: before observing next value we cannot restrict it to a countable set.
• Real-world recognition exercise (C vs D): rainfall, temperature, turtle weight → continuous; houses sold, football score → discrete; eye colour → categorical (not numeric rv).
• Philosophical note: even height in a finite population is technically discrete but modelled as continuous for convenience.

Illustrative ‘truly continuous’ experiment

• Roll a perfect sphere (diameter 2) with a black dot; observe height $H\in[0,2]$ when ball stops. Every value plausible ⇒ H continuous.

Density Curves & Density Functions

• Discrete pmf $p(\cdot)$ ↔ continuous pdf $f(\cdot)$.
• Properties of a valid density:
  • $f(x)\ge 0\,\forall x$ (no negative density).
  • Total area $\int_{-\infty}^{\infty}f(x)dx=1$.
• Probability via area: P(a<X<b)=\int_{a}^{b}f(x)dx (integration not examinable; use geometry/R numerics).
• Histogram ↔ smoothed density (kernel density example: UNSW travel times).

Examples & Exercises

• Uniform[0,2] numbers:
  • Density height $=\frac12$ (rectangle), area = probability.
  • Results: $P(0\le X\le1)=0.5$, $P(X=1/4)=0$, general $P(a\le X\le b)=\dfrac{b-a}{2}$.
• Baby‐smile times assumed $Unif[0,23]$: probability between 2 & 18 = $(18-2)/23$; conditional $P(X\ge12\mid X\ge8)=\frac{11}{15}$.

Mean & Variance for Continuous rv’s (concept only)

• $E(X)=\int x f(x)dx$, $Var(X)=\int (x-\mu)^2 f(x)dx$ – parallels discrete sums.
• Rules for means/variances identical to discrete case.

Lecture 2 – Normal Distributions

Motivation & Central Limit Theorem (CLT)

• Many natural/aggregated measures are approx. normal (sums/averages of many small independent effects).
• CLT preview: means of many independent rv’s tend toward normal distribution irrespective of parent.

Anatomy of a Normal Curve

• Parameters: mean $\mu$ (centre), standard deviation $\sigma$ (spread).
• Formula (not required to memorise): $f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\bigl( -\tfrac{(x-\mu)^2}{2\sigma^{2}} \bigr)$.
• Notation (course uses SD): $X\sim N(\mu,\sigma)$; many texts use variance $N(\mu,\sigma^{2})$.

Normality Assumption & Assessment

• Assess normality before using normal-based methods:
  • Histograms (quick but bin-width sensitive).
  • Normal Quantile (Q-Q) plot (best): plot ordered data vs theoretical quantiles $q_i=\Phi^{-1}\left(\tfrac{i}{n+1}\right)$.
  • If data come from normal, points≈straight line.
  • Systematic curves indicate skewness or heavy tails.
• Boxplots nearly useless for normality.
• Use a scale (excellent → hopeless) rather than yes/no.

68–95–99.7 Empirical Rule

• For normal rv $X\sim N(\mu,\sigma)$:
  • P(|X-\mu|<\sigma)\approx0.68.
  • P(|X-\mu|<2\sigma)\approx0.95.
  • P(|X-\mu|<3\sigma)\approx0.997.
• IQ example: $N(100,15)$. 68 % between 85 & 115; 95 % between 70 & 130; 99.7 % between 55 & 145.

Other Continuous Models Introduced

• Exponential $Expo(\lambda)$: waiting time to first success; mean $1/\lambda$; memoryless.
• Weibull: generalises exponential by allowing hazard rate $\propto t^{k-1}$.
• Gamma $Gamma(a,\lambda)$: sum of $a$ i.i.d. exponential waits; mean $a/\lambda$.

Lecture 3 – Probabilities & Quantiles from a Normal Distribution

Standard Normal $Z$

• Defined as $N(0,1)$; key for all probability work.
• Use R:
  • P(Z<z) ⇒ pnorm(z) (cumulative left-tail).
  • Reverse: find $c$ s.t. P(Z<c)=p ⇒ qnorm(p).

Example Conversions

• P(Z<1.4)=0.9192.
• Two-sided interval: P(-1.39<Z<0.43)=pnorm(0.43)-pnorm(-1.39)=0.5841.
• Right tail: P(Z>1.4)=1-pnorm(1.4)=0.0807 (or lower.tail=FALSE).

Standardising Any Normal $X\sim N(\mu,\sigma)$

• Transform $Z=\dfrac{X-\mu}{\sigma}$ ⇒ $Z\sim N(0,1)$.
• Probability statement converts accordingly.
• IQ example revisited: P(X>110)=P(Z>\tfrac{10}{15})\approx0.2525.
• Birth-weight example: $X\sim N(3500,600)$; P(2000<X<3000)=P(-2.5<Z<-0.833)=0.196.

Using R Arguments vs Standardisation

• Direct: pnorm(upper,mean,sd)-pnorm(lower,mean,sd).
• Via Z: transform bounds then use default pnorm.

Additional Practice

• Pregnancy length (266 ± 16 days) with 68–95–99.7 rule.
• Find z-values for given cumulative/tail probabilities using qnorm.

Inside the Normal Quantile Plot (mechanics)

• For sample size $n$, theoretical quantiles: $q_i=\Phi^{-1}\bigl(\tfrac{i}{n+1}\bigr)$.
• Plot pairs $(q<em>i,x</em>{(i)})$; linearity ↔ normality.

Lecture 4 – Simulation Examples

Pseudo-Random Number Generation

• Computers produce pseudo-random numbers via deterministic algorithms + seed.
• Linear Congruential Generator (illustrated):
  • Modulus $m=2^{31}-1$; multiplier 48271.
  • Recurrence $x<em>{j+1}=(48271 x</em>j)\;\text{mod}\; m$ ⇒ scale to $[0,1]$.
• Implemented as my.runif(); compared with R’s runif() (faster, higher quality).

From Uniform to Normal – Box–Muller

• If $u<em>1,u</em>2\sim Unif[0,1]$ independent:
  • $z<em>1=\sqrt{-2\ln u</em>1}\cos(2\pi u_2)$,
  • $z<em>2=\sqrt{-2\ln u</em>1}\sin(2\pi u_2)$ ⇒ i.i.d. $N(0,1)$.
• Encapsulated in my.rnorm(); validated by large-sample mean ≈ 0, variance ≈ 1 and histogram overlapped with dnorm.

R Function Naming Conventions

• rname → random generation; dname → density; pname → CDF; qname → quantile.
• Examples: runif, rnorm, rchisq, rt, rf.
• Extra distributions accessible via package PoweR (40+ laws).

Simulation for Unknown/Complex Distributions

• Core idea: if model can be simulated, any quantity (mean, SD, prob.) approximated by Monte Carlo:
  • Generate large $M$.
  • Estimate $E[g(X)]\approx\tfrac1M\sum<em>{i=1}^{M}g(x</em>i)$.
  • Probability $P(A)=E[1_A(X)]\approx\text{mean}(\text{indicator})$.
• Demonstration:
  • If $X\sim N(2,7)$, let $Y=X^2$. Monte Carlo with 100k draws yielded $E(Y)\approx53$, $SD(Y)\approx75$, P(Y>5)\approx0.759.

Ant-Colony Case Study (interaction probability)

• Model: positions $C<em>1:(X</em>{1},Y<em>{1})\sim N(0,30)\times N(0,30)$; $C</em>2:(X<em>{2},Y</em>{2})\sim N(100,30)\times N(100,30)$; independence.
• Distance between random ants: $D=\sqrt{(X<em>1-X</em>2)^2+(Y<em>1-Y</em>2)^2}$.
• Simulate $M=10^5$ pairs: mean(D<=70)=0.0291 ⇒ ≈2.9 % of pairs within 70 m (half-way) → “very little interaction”.

Success Criteria Recap

• Able to:
  • Recognise continuous vs discrete.
  • Interpret & sketch density curves.
  • Apply normality assumption checks (histogram, Q-Q).
  • Use pnorm, qnorm for probability & quantile tasks.
  • Standardise vs supply (mean,sd).
  • Generate pseudo-random numbers & use Monte Carlo for arbitrary probs.

Ethical, Philosophical & Practical Implications

• Modelling choice (continuous vs discrete) often pragmatic; must be justified by analysis goal and measurement precision.
• Over-reliance on normality dangerous; always assess – “statistics is not exact, think critically rather than follow recipes”.
• Simulation allows risk-free exploration (e.g. nuclear test modelling) but quality hinges on generator quality and valid underlying assumptions.

Key R Syntax Cheat-Sheet

• pnorm(z,mean=μ,sd=σ,lower.tail=FALSE) → P(X>z).
• qnorm(p) → $z<em>p$ such that $P(Z<z</em>p)=p$.
• runif(n,min,max) | rnorm(n,mean,sd) → generate.
• Probability via simulation: mean(expr) where expr is logical vector.
• Histogram with density: hist(x,prob=TRUE); curve(dname(...),add=TRUE).
• Q-Q plot: qqnorm(x); qqline(x).

Terminology (Keywords Slide)

• probability model; density curve/function; normal curve; z-score; seed; pseudo-random number; quantile; lower.tail; Monte Carlo; memoryless property.