102 Session 15

102 Session15 Random rates

© KSES Exam questions are copyright Faculty & Institute of Actuaries & are used with their permission Source: www.actuaries.org.uk 432

Compare these samples (state the obvious)

433

Compare these samples (state the obvious)

Circles are “higher” on average ie they have higher mean 434

Compare these samples

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Compare these samples

Circles are more spread on average. Circles vary more 436

Compare these pairs

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Compare these pairs

(Circles have same mean and variance in both pairs.) The second pair co-varies more – if one rises, the other tends to rise too 438

Define terms Expectation = average value (weighted by probability) Variance

= average spread (weighted by probability) = expectation of (squared) spread

Covariance = average paired-spread (weighted by prob.) = expectation of paired-spread

439

Random definitions Define, and explain as if to a seven-year-old: 1. Expectation. 2. Variance 3. Covariance 4. Independence Explain why the definitions are the way they are

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Random results Prove the following (from their definitions) and give an illustration of each E(A + B) = E(A) + E(B) V(A) := Var(A) = E(A2) – E(A)2 Cov(A,B) = E(AB) – E(A)E(B) A,B independent => E(AB) = E(A)E(B) Var(kA) = k2 V(A) Var(A+B) = Var(A) + Var(B) + 2cov(A,B) A,B indep. => Var(A+B) = Var(A) + Var(B)441

Most useful results You’ll probably use the bold results most often in random rates questions: E(A + B) = E(A) + E(B) V(A) := Var(A) = E(A2) – E(A)2 Cov(A,B) = E(AB) – E(A)E(B) A,B independent => E(AB) = E(A)E(B) Var(kA) = k2 V(A) Var(A+B) = Var(A) + Var(B) + 2cov(A,B) 442 A,B indep. => Var(A+B) = Var(A) + Var(B)

Apr 2003 Q9

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Apr 2003 Q9

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Specimen Q7

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Specimen Q7

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Sep 2002 Q8

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Sep 2003 Q9

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Sep 2003 Q9(i)

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Sep 2003 Q9(ii)

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Apr 2002 Q11(i)

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Apr 2002 Q11(i)

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LogNormal definition

If X ~ LogN(μ,σ2) How is Log(X) distributed?

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LogNormal mean If X ~ LogN(μ,σ2) then accept or prove that E(X) = e

μ+ ½ σ2

What is E(log X)? What is log(E(X)) ? Does E(log X) = log(E(X)) ? 454

LogNormal variance If X ~ LogN(μ,σ2) then accept or prove that Var(X) = = =

2 2μ+ σ (e )

2 σ (e -1)

2 2 σ Mean (e -1)

2 σ e

Mean2 -Mean2 455

Finding LogNormal parameters If X ~ LogN(μ,σ2) Var(X) =

2 2 σ Mean (e -1)

Mean = e

μ+ ½ σ2

So given the mean and variance, 2 σ calculate (e -1), then σ, then μ 456

Specimen Q14 (i)

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Specimen Q14 (i)

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Apr 2000 Q11(i)

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Apr 2000 Q11(i)

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Sum of LogNormals

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You need to accept or prove that …

If you … add any normal distribution to any

normal distribution, you get a …

normal distribution

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Normal result Assume that if you add two normal distributions, the result is a normal distribution. If X ~ N(μ,σ2) and Y~N(m,s2) How is X + Y distributed? 463

Apr 2001 Q9(i)

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Apr 2001 Q9(i)

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Apr 2000 Q11(ii)a

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Apr 2000 Q11(ii)a

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Normal distribution function

Find the Tables for the Standard Normal distribution function. If X~N(0,1) then Probability than X < c =: Φ(c) 468

Using Normal distribution If X~N(4,25) then X – 4 ~ N(0,25) and (X – 4)/5 ~ N(0,1) So Probability that X < 14 = Probability that X – 4 < 14 – 4 = Probability that (X – 4)/5 < (14 – 4)/5= 2 = Probability that N(0,1) < 2 =

Φ(2)

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Sep 2003 Q5

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Sep 2003 Q5

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Specimen Q14 (ii)

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Specimen Q14 (ii)

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Specimen Q14 (ii)

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Apr 2000 Q11(ii)b

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Apr 2000 Q11(ii)b

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Apr 2002 Q11(ii)&(iii)

477

Apr 2001 Q9(ii)

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Sep 2000 Q8

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Sep 2000 Q8

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Sep 2001 Q6

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Sep 2001 Q6

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END of 102 sessions See also examiners’ comments on rounding

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