Question

A box contains four slips of paper numbered 1, 2, 3 and 4. You are to select 2 slips without replacement. Consider the random variables:

M = the maximum of the two slips,

D = the absolute difference between the slips.

P_1= payout 1 = M -YD + 2

P_2= payout 2=MD

a) What is the expected value of payout 2?

b)If M and D are as expected and Y ~ Uniform(0, a) , what is the probability payout 1 is positive?

Answer 1

a) It would be best to tabulate all possible combinations of the two slips, compute the corresponding values of M and D. The product of these 2 values is the value of Payout 2. Note that the value on two slips cannot be equal, since they are chosen without replacement. Hence,

#	Slip 1	Slip 2	M	D	MD
1	1	2	2	1	2
2	1	3	3	2	6
3	1	4	4	3	12
4	2	1	2	1	2
5	2	3	3	1	3
6	2	4	4	2	8
7	3	1	3	2	6
8	3	2	3	1	3
9	3	4	4	1	4
10	4	1	4	3	12
11	4	2	4	2	8
12	4	3	4	1	4
			SUM = 40	SUM = 20	SUM = 70

The sum of all possible values of Payout 2 comes to be 70. Since the slips are chosen independently, hence all 12 combinations are equally likely. Thus, the expected value is

b) In the same table, we need to compute the value of Payout 1. In order to find expected value, we note that

From table 1, we already have the sum of M and D, from which we get