Question

Use the frequency and severity distributions given to answer the following questions.

Frequency Distribution		Severity Distribution
Losses	Probability	Loss Amount	Probability
0	40%	$400	20%
1	35%	$700	45%
2	25%	$900	35%

Mean=? Mean=$710

1. Calculate the mean of the frequency distribution.

2.Determined the total loss distribution. Show all of the possible losses and their associated probabilities. Prove that the probabilities add to one.

3. Calculate the mean of the total loss distribution and show that is equal to the mean of the severity distribution times the mean of the frequency distribution.

Answer 1

1.

Mean of the frequency distribution is

Losses (f)	Probability
0	0.40	0 * 0.40 = 0.00
1	0.35	1 * 0.35 = 0.35
2	0.25	2 * 0.25 = 0.50
Total	1.00	0.85

Therefore mean of the frequency distribution =

2.

The total loss distribution L (=frequency * Severity) with possible losses and their associated probabilities are given below in the table:

frequency (f)	Severity (s)	Loss L = (f*s)	Probability p(L)
0	400	0	0.08
0	700	0	0.18
0	900	0	0.14
1	400	400	0.07
1	700	700	0.16
1	900	900	0.12
2	400	800	0.05
2	700	1400	0.11
2	900	1800	0.09
Total			1.00

Total probabilities of all possible losses

3.

Mean of the total Loss distribution from the Loss distribution table is

.. (I)

From (1)

  

... (II)

From (I) and (II)

Mean of the total loss distribution, is equal to the mean of the severity distribution times the mean of the frequency distribution