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May 13'23
Exercise
The distribution of a loss, [math]X[/math], is a two-point mixture:
- With probability 0.8, [math]X[/math] has a Pareto distribution with [math]\alpha = 2[/math] and [math]\theta = 100[/math].
- With probability 0.2, [math]X[/math] has a Pareto distribution with with [math]\alpha = 4[/math] and [math]\theta = 3000 [/math].
Calculate [math]\operatorname{P}( X \leq 200).[/math]
- 0.76
- 0.79
- 0.82
- 0.85
- 0.88
Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.
May 13'23
Key: A
[[math]]
F (200) = 0.8 \left [ 1-\left( \frac{100}{200 +100}\right)^2\right] + 0.2 \left [ 1 - \left( \frac{3000}{3000 +200}\right)^4\right ] = 0.7566
[[/math]]
Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.