Question

Richard is a German exchange student at Iowa. He is al- ways in a hurry and forgets to pay for parking. The probability of getting a parking ticket is 5% and the fine is $40.

a. Find mean, variance and skewness of his fine payment.

Peter is another German exchange student at Iowa with exactly the same problem, a 5% probability of having to pay a fine of $40. His risk of getting a parking ticket is independent of Richard’s. They decide to pool their risks. Each of them pays half of the total fines.

b. Determine the probability distribution of Richard’s fine payment under the pooling arrangement.

c. Find mean, variance and skewness of Richard’s fine payment under the pooling ar- rangement. Compare a. to c. What is the effect of pooling?

Answer 1

(a)

Let X denote the random variable representing Richard's fine payment (in $). Now we are given the following info:

P(X=40) = 0.05

P(X=0) = 1 - 0.05 = 0.95

Now, the mean of fine payment:

For Variance, we first calculate:

Thus, variance:

For skewness we first calculate:

Thus, skewness:

(b)

Let Y denote the random variable representing the fine paid by Richard under the pooling arrangement.

The possible values of Y are 0(if no one gets a parking ticket), 20(when one of them gets a parking ticket) and 40(when both get a parking ticket)

Now, P(Y=0) = P(Richard doesnt get a parking ticket)*P(Peter doesnt get a parking ticket)

= (1-0.05)*(1-0.05)

= 0.9025

P(Y=20) = P(Richard gets a ticket)*P(Peter doesnt get a ticket) + P(Richard doesnt get a ticket)*P(Peter gets a ticket)

= 0.05*(1-0.05) + (1-0.05)*0.05

= 0.05*0.95 + 0.05*0.95

= 0.095

P(Y=40) = P(Richard gets a parking ticket)*P(Peter gets a parking ticket)

= 0.05*0.05

= 0.0025

Thus, the distribution of Richard's fine payment under pooling arrangement is given by:

y	P(Y=y)
0	0.9025
20	0.0950
40	0.0025

(c)

We first find the first three moments about origin:

Thus, mean:

We observe that :

Mean(X) = Mean(Y)

Var(X) > Var(Y)

Skew(X) > Skew(Y)

Thus, pooling doesn't have any effect on the mean but reduces the variance and skewness of fine payments.

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