1.

Verify (4.72), (4.73), (4.75)/(4.76), and (4.80). Solve ? and ? from (4.80).

2.

Work out the details of the final approximation.

1.

.

2.

p _n( t + dt) = p _n( t) + ( t) dt = (1 - ? dt) p _n( t) + ? dtp _{n- 1}( t). Both sides denote the probability of n, + 1 claims in (0, t + dt).

1.

Prove that the expressions in (4.11) are indeed equivalent to (4.10).

2.

Use e ^rx > 1 + rx + ( rx) ² for r > 0 and x > 0 to prove that R < 2 ? ? ₁/ ? ₂.

3.

For ? = 0.4 and p( x) = (3 e ^{-3 x} + 7 e ^{-7 x}), determine the values of t for which m _X( t) is finite, and also determine R.

4.

If the claims distribution is discrete with p(1) = p(2) = , then determine ? if it is given that R = log 3.

5.

Which premium yields e ^{- Ru} = ??

6.

[ ?] If Pr[ X = 0, 1, 2, 4] = , then determine R by using a spreadsheet, for