Modern Actuarial Risk Theory

Tables

Table A: Continuous distributions

Distribution

Density

Parameters, range

Moments, cumulants

Mgf

Uniform( a, b)

a < x < b

E = ( a + b)/2,

Var = ( b - a) 2/12,

? = 0

Normal( ?, ? 2)

? > 0

E = ?, Var = ? 2,

? = 0 ( ? j = 0, j ? 3)

Gamma( ?, ?)

? > 0, ? > 0

x > 0

E = ?/ ?,

Var = ?/ ? 2,

? = 2 ? -1/2

Exponential( ?)

? Gamma(1, ?)

X 2( k)

? Gamma( k/2,1/2)

Inverse Gaussian( ?, ?)

? > 0, ? > 0

x > 0

E = ?/ ?,

Var = ?/ ? 2

? =3 ? -1/2

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

,

Table B: Discrete distributions

Distribution

Density

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