The distribution of a loss, [math]X[/math], is a two-point mixture:

1. With probability 0.8, [math]X[/math] has a Pareto distribution with [math]\alpha = 2[/math] and [math]\theta = 100[/math].
2. 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.

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.