Question

There are 6 numbered balls in a bag. Each ball has a distinct number and the numbers are in {1, 2, 3, 4, 5, 6}. Take 3 balls from the bag (without replacement) randomly and read the number on each ball. Let X1 be the maximum number and X2 be the minimum number among the three observed numbers. (a) Find the marginal p.m.f. of X1. (b) Find the marginal p.m.f. of X2. (c) Find the joint p.d.f. of X1 and X2. (d) Are X1 and X2 independent? Why? (e) Find the correlation coefficient between X1 and X2.

Answer 1

Three balls are choosen out of six balls. The number of choices will be . X1 will be from and X2 will be from .

(a) The marginal pmf of X1 would be as below.

X1	P(X1)
3	1/20
4	3/20
5	6/20
6	10/20

The pmf is derived as below:

• X1=3 when arrives. Hence, there is only 1 way and P(X1=3) is 1/20.
• X1=4 when arrives, and hence there are only 3 ways, and P(X1=4) is 3/20.
• X1=5 when arrives, and hence there are only 6 ways, and P(X1=5) is 6/20.
• X1=6 when arrives, and hence there are 10 ways, and P(X1=6) is 10/20.

(b) The marginal pmf of X2 would be as below.

X2	P(X2)
1	10/20
2	6/20
3	3/20
4	1/20

The pmf of X2 is derived as below (the reverse order is there show the analogy to the previous one):

• X2=4 when arrives and there is only 1 way, and P(X2=4) is 1/20.
• X2=3 when arrives, and hence there are only 3 ways, and P(X2=3) is 3/20.
• X2=2 when arrives, and hence there are only 6 ways, and P(X2=2) is 6/20.
• X2=1 when arrives, and hence there are 10 ways, and P(X2=1) is 10/20.

(c) The joint pmf would be as below (pmf since when the random variables are discrete, the term 'mass' is used, not 'density').

X1/X2	1	2	3	4
3	1/20	0/20	0/20	0/20
4	2/20	1/20	0/20	0/20
5	3/20	2/20	1/20	0/20
6	4/20	3/20	2/20	1/20

The joint pmf is derived as . Now, or or . The rest is derived as below.

• or or
• or or .
• or or .
• or or .
• or or .
• or or .
• or or .
• or or .
• or or .

(d) X1 and X2 would be independent if for all values of X1 and X2. We have , but , and (atleast) as in this one case, we have . Thus, the random variables X1 and X2 are not independent.

(e) The correlation coefficient between two random variables is , where sigma-1 and sigma-2 are standard deviation of X1 and X2.

The mean of X1 is or or or ; while the mean of X2 is or or or .

The table required for rest of the calculation is below.

-2.25	5.0625	-0.75	0.5625	1.6875
-1.25	1.5625	0.25	0.0625	-0.3125
-0.25	0.0625	1.25	1.5625	-0.3125
0.75	0.5625	2.25	5.0625	1.6875

We have, or . Also, or or and   or or . The correlation will be hence   or , which is the correlation coefficient between X1 and X2.