Credibility

In: Business and Management

Submitted By TianjiaoWang
Words 887
Pages 4
Credibility Theory
Lecture Solutions – Week 1

1.1
Let

A be the event that a student is an actuarial student.
C be the event that a student is a Credibility Theory student.

P(C)

= 0.02



P( C ) = 0.98

P(A | C) = 0.98
P(A | C ) = 0.07

P( A | C ) P(C )
P ( A | C ) P(C ) + P( A | C ) P(C )

P (C | A) =
=

0.98 × 0.02
0.98 × 0.02 + 0.07 × 0.98

=

0.0196
0.0882

= 0.2222

= 22%

1.2
Let

A1 = “The science department is a heavy polluter”.
A2 = “The science department is a light polluter”.
A3 = “The science department is not a polluter”.
B = “The waterways are visibly polluted”.

P(A1) = 0.1;

P(A2) = 0.2;

P(A3) = 0.7.

P(B |A1) = 0.9;

P(B |A2) = 0.6;

P(B |A3) = 0.3.

P(B)

= P(B |A1)P(A1) + P(B |A2)P(A2) + P(B |A3)P(A3)
= 0.9*0.1 + 0.6*0.2 + 0.3*0.7
= 0.42

P ( A1 | B) =

P( B | A1 ) P( A1 ) 0.9 × 0.1
=
= 0.21
P( B)
0.42

P ( A2 | B) =

P( B | A2 ) P( A2 ) 0.6 × 0.2
=
= 0.29
P( B)
0.42

P( A3 | B) =

P( B | A3 ) P( A3 ) 0.3 × 0.7
=
= 0.5
P( B)
0.42

1.3
The Bayesian model is: p ~ U(0,1) and (X|p) ~ Bin(n,p).
Thus, f(p) = 1, 0 ≤ p ≤ 1

and

⎛n⎞ f(x|p) = ⎜ ⎟ p x (1 − p) n − x .
⎝x⎠

⎛n⎞ x n −x
⎜ ⎟ p (1 − p) x f(p|x) = ⎝ ⎠
1⎛ n ⎞ x n −x
∫0 ⎜ x ⎟ p (1 − p) dp
⎝ ⎠
=

p x (1 − p) n − x
1

∫p
0

Recall:

1

∫p
0

α−1

x

(1 − p) n − x dp

(1 − p)β−1 dp =

Γ(α)Γ(β)
Γ(α + β)

Therefore, f(p|x) =

Γ(n + 2) p x (1 − p) n − x ,
Γ(x + 1)Γ(n − x + 1)

0≤p≤1

(p|x) ~ Beta(x+1, n – x + 1)

1.4
Let

X be an indicator variable such that X = 1 if a card drawn from the urn (with replacement) is observed to be white and X = 0, otherwise.
W be the number of white cards in the urn.

Given p, X is a Bernoulli random variable with parameter p.

Since we have not been given any information regarding the distribution of the number of…...

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