Chapter 3 – Linear Combinations of Random Variables

Overview

• Chapter focus: Linear combinations of random variables and their impact on expectation (mean), variance and probability calculations.
• Why it matters:
  • Real-world metrics often aggregate several independent random quantities (e.g. monthly stock portfolio profit, triathlon time, weight of jars of honey).
  • Understanding linear combinations lets us derive overall distributions and associated probabilities instead of analysing each component separately.

Expectation & Variance for Translating/Scaling One Random Variable

• Translation by constant $b$
  • $E\,(X+b)=E\,(X)+b$
  • $\operatorname{Var}\,(X+b)=\operatorname{Var}\,(X)$
• Scaling by constant $a$
  • $E\,(aX)=a\,E\,(X)$
  • $\operatorname{Var}\,(aX)=a^{2}\,\operatorname{Var}\,(X)$
• Combined rule (Key Point 3.3):
  $E\,(aX+b)=a\,E\,(X)+b$
  $\operatorname{Var}\,(aX+b)=a^{2}\,\operatorname{Var}\,(X)$

Illustrative Dice Examples (Section 3.1)

• Xing’s die values: ${1,1,2,2,2,4}$
  • $E\,(X)=2$; $\operatorname{Var}\,(X)=1$
• Yaffa’s die values: ${4,4,5,5,5,7}=X+3$
  • $E\,(Y)=E\,(X)+3=5$
  • $\operatorname{Var}\,(Y)=\operatorname{Var}\,(X)=1$ (shift does not alter variance)
• Quenby’s die values: ${2,2,4,4,4,8}=2X$
  • $E\,(Q)=2E\,(X)=4$
  • $\operatorname{Var}\,(Q)=4\operatorname{Var}\,(X)=4$ (variance quadruples because $a^{2}=4$)
• Exploratory Tasks: add/subtract constants to Mo’s die ${0,0,1,1,1,3}$ or create a rule like $3X+1$ to confirm general results.

Worked Example 3.1 (Table → 2X+3)

• Given discrete $P(X)$ table, first compute raw $E(X)=4$ and $\operatorname{Var}(X)=3.8$.
  • Then: $E(2X+3)=2\times4+3=11$
  • $\operatorname{Var}(2X+3)=4\times3.8=15.2$

Binomial Link (Worked Example 3.2)

• For $X\sim B(3,\tfrac12)$ : $E(X)=1.5$, $\operatorname{Var}(X)=0.75$.
• Same rules apply: $E(2X+1)=2(1.5)+1=4$, $\operatorname{Var}(2X+1)=4(0.75)=3$.

Linear Combinations of Two Independent RVs (Key Point 3.4)

• For independent $X,\,Y$:
  $E\,(X+Y)=E\,(X)+E\,(Y)$
  $\operatorname{Var}\,(X+Y)=\operatorname{Var}\,(X)+\operatorname{Var}\,(Y)$
• Difference: $E\,(X-Y)=E\,(X)-E\,(Y)$ but variance still adds:
  $\operatorname{Var}\,(X-Y)=\operatorname{Var}\,(X)+\operatorname{Var}\,(Y)$

Multiples Before Summation (Key Point 3.5)

• For constants $a,b$ and independent $X,Y$:
  $E\,(aX+bY)=aE\,(X)+bE\,(Y)$
  $\operatorname{Var}\,(aX+bY)=a^{2}\operatorname{Var}\,(X)+b^{2}\operatorname{Var}\,(Y)$
• Extends to any finite sum of independent variables.

Dice Illustration (Green vs Blue tetrahedral dice)

• Green $G$: $E=\tfrac43$, $\operatorname{Var}=\tfrac{11}{16}$
• Blue $B$: $E=\tfrac32$, $\operatorname{Var}=\tfrac14$
• Sum $W=G+B$ → $E(W)=\tfrac14+\tfrac32=\tfrac{7}{4}$, $\operatorname{Var}(W)=\tfrac{15}{16}$ (matches explicit enumerated distribution).

Scaled Combination Example

• If we redefine $D=2G+3B$ then:
  • $E(D)=2E(G)+3E(B)=2(\tfrac43)+3(\tfrac32)=8$
  • $\operatorname{Var}(D)=4\,\tfrac{11}{16}+9\,\tfrac14=5$ (again matches table).

Distinguishing $2X$ vs $X<em>1+X</em>2$

• $2X$: double a single observation (same sample) → variance $4\operatorname{Var}(X)$.
• $X<em>1+X</em>2$: sum of two independent observations → variance $2\operatorname{Var}(X)$.
• Example with Xing’s die validates:
  $E(2X)=4$, $\operatorname{Var}(2X)=4$ vs
  $E(X<em>1+X</em>2)=4$, $\operatorname{Var}(X<em>1+X</em>2)=2$.

Normal Distributions (Key Point 3.6)

• If $X\sim N(\mu,\sigma^{2})$ then any linear form $aX+b\sim N(a\mu+b,\,a^{2}\sigma^{2})$.
• If independent $X\sim N(\mu<em>1,\sigma</em>1^{2})$ and $Y\sim N(\mu<em>2,\sigma</em>2^{2})$ then $aX+bY\sim N(a\mu<em>1+b\mu</em>2,\,a^{2}\sigma<em>1^{2}+b^{2}\sigma</em>2^{2})$.

Normal Examples

1. Four Thrift batteries $T\sim N(7,2.3^{2})$ → Sum $S$ has
   $E(S)=28$, $\operatorname{Var}(S)=4(2.3^{2})=21.16$, $S\sim N(28,21.16)$.
   Probability $P(S>30)=0.332$.
2. Large vs small rice bags: $Y\sim N(6.6,0.4^{2})$, $X\sim N(2.1,0.2^{2})$.
   Want $P(Y>3X)$. Define $Z=Y-3X\sim N(0.3,0.52)$ then $P(Z>0)=0.661$.
3. Worktop thickness:
   • Top only: $37+1\Rightarrow N(38,0.09)$.
   • Top & bottom: $37+1+1\Rightarrow N(39,0.0902)$.
4. Gift package: Total mass $3S+T\sim N(240,13^{2})$, cheap-rate probability $\approx0.779$ for mass <250 g.

Poisson Combinations (Key Point 3.7)

• Independent $X\sim \operatorname{Po}(\lambda)$, $Y\sim \operatorname{Po}(\mu)$ ⇒ $X+Y\sim \operatorname{Po}(\lambda+\mu)$.
• Important: only sums preserve Poisson; differences or scaling do not stay Poisson.

Rescue-centre Story

• Lions $L\sim\operatorname{Po}(5)$, Tigers $T\sim\operatorname{Po}(3)$ → Total $A=L+T\sim\operatorname{Po}(8)$.
• Probability of rescuing exactly 2 animals: $P(A=2)=e^{-8}\dfrac{8^{2}}{2!}=0.0107$ (far quicker than enumerating each lion/tiger combination).

Text-message Example

• Josh $\lambda=3.2$, Reuben $\lambda=2.5$ ⇒ $T\sim\operatorname{Po}(5.7)$.
  $P(T>5)=1-P(T\le4)=0.327$.

Non-Poisson after Linear Ops

• If $T=2X-Y$ where $X\sim\operatorname{Po}(2.4),\,Y\sim\operatorname{Po}(3.6)$:
  • $E(T)=1.2$, $\operatorname{Var}(T)=13.2$.
  • Since mean ≠ variance, $T$ is not Poisson.

Ethical/Practical Implications & Real-World Links

• Finance: Portfolio profit as sum of independent share returns.
• Manufacturing: Jar of honey weight = jar + honey + lid; tolerance analysis uses variance formulas.
• Sports: Triathlon or relay times are sums of event times; probability of finishing under target uses normal combination.
• Quality control: Worktop thickness, rice packages, and soaps rely on aggregated normal models for compliance thresholds.
• Communication planning: Text-message Poisson combo informs network capacity.

Connections to Prior Learning

• Relies on discrete expectation/variance (P&S 1 Chapters 6–7).
• Uses Binomial, Poisson, Normal models from earlier chapters and coursebooks.
• Standardisation $z=\dfrac{x-\mu}{\sigma}$ remains foundational for probability lookup.

Formulae Checklist (end-of-chapter summary)

• Single RV, constants $a,b$:
  • $E(aX+b)=aE(X)+b$
  • $\operatorname{Var}(aX+b)=a^{2}\operatorname{Var}(X)$
• Independent $X,Y$:
  • $E(aX+bY)=aE(X)+bE(Y)$
  • $\operatorname{Var}(aX+bY)=a^{2}\operatorname{Var}(X)+b^{2}\operatorname{Var}(Y)$
• Normal closure:
  • $X\sim N(\mu,\sigma^{2})\Rightarrow aX+b\sim N(a\mu+b,a^{2}\sigma^{2})$
  • $X,Y$ independent normals ⇒ $aX+bY$ also normal.
• Poisson closure:
  • $X\sim \operatorname{Po}(\lambda),\,Y\sim \operatorname{Po}(\mu)$ (independent) ⇒ $X+Y\sim \operatorname{Po}(\lambda+\mu)$.

Worked-Example Shortcuts & Tips

• When summing n identical independent RVs $X$:
  $E\,(\text{sum})=nE(X)$, $\operatorname{Var}(\text{sum})=n\operatorname{Var}(X)$.
• For differences, variance always adds: $\operatorname{Var}(X-Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$.
• Use standardisation for normal probabilities:
  $P(X>k)=1-\Phi\Big(\dfrac{k-\mu}{\sigma}\Big)$.
• For Poisson ‘greater than’, compute complement with cumulative sum of small $k$ values to reduce computations.

Worked & Exercise References

• Exercise sets 3A–3D reinforce computing expectations & variances, distinguishing $2X$ vs $X<em>1+X</em>2$, normal combination probabilities, Poisson sums, and practical conversions (°C→°F).
• End-of-chapter review questions apply concepts to temperatures, egg boxes, triathlon times, cycling hire cost, mining value, etc.