Modern Actuarial Risk Theory

| Distribution | Density | Parameters, range | Moments, cumulants | Mgf |
|---|---|---|---|---|
| Uniform( a, b) |
![]() | a < x < b | E = ( a + b)/2, Var = ( b - a) 2/12, ? = 0 |
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| Normal( ?, ? 2) |
![]() | ? > 0 | E = ?, Var = ? 2, ? = 0 ( ? j = 0, j ? 3) |
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| Gamma( ?, ?) |
![]() | ? > 0, ? > 0 x > 0 | E = ?/ ?, Var = ?/ ? 2, ? = 2 ? -1/2 |
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| Exponential( ?) | ? Gamma(1, ?) | |||
| X 2( k) | ? Gamma( k/2,1/2) | |||
| Inverse Gaussian( ?, ?) |
![]() | ? > 0, ? > 0 x > 0 | E = ?/ ?, Var = ?/ ? 2 ? =3 ? -1/2 |
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| Beta( a, b) |
![]() | a > 0, b > 0 0 < x < 1 | | |
| Lognormal( ?, ? 2) |
![]() | ? > 0 x > 0 | ? = c 3 + 3 c with c 2 = Var/E 2 | |
| Pareto( ?, x 0) |
| ? > 0, x > x 0 > 0 | |
| Distribution | Density |
|---|