Porisma

Worked solution

Take fourth moments when an area is proportional to the square

S has distribution

sP(S=s)P(S=s)
10.5
50.3
100.1
200.1

A circle has radius R=2SR=2S.

(i) Find P(S(11−S)≤10)P(S(11-S)\le10). marks

(ii) Find E(S)E(S) and Var⁡(S)\operatorname{Var}(S). marks

(iii) Hence find the mean and variance of the circle’s area A. marks

[2006/MJC/II/26]

CJC · Review problems · Question 7

Answer

(i)0.7; (ii)E(S)=5E(S)=5, variance 33; (iii)E(A)=232πE(A)=232\pi, variance 221184π2221184\pi^2.

(i) Retain the factorisation and both boundary scores

P(S(11−S)≤10)=P(S2−11S+10≥0)=P((S−1)(S−10)≥0)=P(S≤1 or S≥10)=P(S=1)+P(S=10)+P(S=20)=0.5+0.1+0.1=0.7P(S(11-S)\le10)=P(S^2-11S+10\ge0)=P((S-1)(S-10)\ge0)=P(S\le1\text{ or }S\ge10)=P(S=1)+P(S=10)+P(S=20)=0.5+0.1+0.1=0.7. The source’s next line prints P(S>10)P(S>10), but its included score 10 and the original non-strict inequality require P(S≥10)P(S\ge10).

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(ii) Mean, second moment and variance

E(S)=1(0.5)+5(0.3)+10(0.1)+20(0.1)=5E(S)=1(0.5)+5(0.3)+10(0.1)+20(0.1)=5. E(S2)=12(0.5)+52(0.3)+102(0.1)+202(0.1)=58E(S^2)=1^2(0.5)+5^2(0.3)+10^2(0.1)+20^2(0.1)=58. Therefore Var⁡(S)=58−52=33\operatorname{Var}(S)=58-5^2=33.

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(iii) Area and mean

A=πR2=4πS2A=\pi R^2=4\pi S^2. Hence E(A)=E(4πS2)=4πE(S2)=4π(58)=232πE(A)=E(4\pi S^2)=4\pi E(S^2)=4\pi(58)=232\pi.

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(iii) Retain the fourth moment and variance of the square

Var⁡(A)=Var⁡(4πS2)=16π2Var⁡(S2)\operatorname{Var}(A)=\operatorname{Var}(4\pi S^2)=16\pi^2\operatorname{Var}(S^2). Now E(S4)=14(0.5)+54(0.3)+104(0.1)+204(0.1)=17188E(S^4)=1^4(0.5)+5^4(0.3)+10^4(0.1)+20^4(0.1)=17188. Thus Var⁡(S2)=E[(S2)2]−[E(S2)]2=E(S4)−582=17188−582=13824\operatorname{Var}(S^2)=E[(S^2)^2]-[E(S^2)]^2=E(S^4)-58^2=17188-58^2=13824.

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(iii) Apply the scale factor squared

Var⁡(A)=16π2Var⁡(S2)=16π2(13824)=221184π2\operatorname{Var}(A)=16\pi^2\operatorname{Var}(S^2)=16\pi^2(13824)=221184\pi^2.

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Common mistakes

Var(S²) is not [Var(S)]². The qualifying boundary score 10 must remain included.

Solution notes

All asks/marks and table entries retained; every source factorisation, moment/variance term and area transformation retained. Source strict-inequality slip corrected explicitly to include 10, native original preserved.

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