In: Statistics and Probability
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.
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:
(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):
(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.
(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.
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-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 |
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We have,
or
. Also,
or
or
and
or
or
. The correlation will be hence
or
, which is the correlation coefficient between X1 and X2.