Question

In: Statistics and Probability

You roll two fair dice, and denote the number they show by X and Y. Let...

You roll two fair dice, and denote the number they show by X and Y. Let U = min{X, Y } and V = max{X, Y }. Write down the joint probability mass function of (U, V ) and compute ρ(U, V ) i.e the correlation coefficient of U and V

Expert Solution

Following is the sample space of the rolling of two fair dice:

U shows the minimum of two dice. For example: for outcome (1,2) , U is 1.

V shows the maximum of two dice. For example: for outcome (1,2) , V is 2.

Since there are two outcome (1,2) and (2,1) for which minimum is 1 and maximum is 2. So

P(U=1, V=2) = 2/36

Likewise following table shows the joint pdf of U and V:

					U
		1	2	3	4	5	6	P(V=v)
	1	1/36	0	0	0	0	0	1/36
	2	2/36	1/36	0	0	0	0	3/36
V	3	2/36	2/36	1/36	0	0	0	5/36
	4	2/36	2/36	2/36	1/36	0	0	7/36
	5	2/36	2/36	2/36	2/36	1/36	0	9/36
	6	2/36	2/36	2/36	2/36	2/36	1/36	11/36
	P(U=u)	11/36	9/36	7/36	5/36	3/36	1/36	1

---------------------------

Now we need to find the mean and variance of U and V and E(UV). Following table shows the calculations for E(UV):

U	V	P(U=u, V=v)	UVP(U=u, V=v)
1	1	1/36	1/36
1	2	2/36	4/36
1	3	2/36	6/36
1	4	2/36	8/36
1	5	2/36	10/36
1	6	2/36	12/36
2	1	0	0
2	2	1/36	4/36
2	3	2/36	12/36
2	4	2/36	16/36
2	5	2/36	20/36
2	6	2/36	24/36
3	1	0	0
3	2	0	0
3	3	1/36	9/36
3	4	2/36	24/36
3	5	2/36	30/36
3	6	2/36	36/36
4	1	0	0
4	2	0	0
4	3	0	0
4	4	1/36	16/36
4	5	2/36	40/36
4	6	2/36	48/36
5	1	0	0
5	2	0	0
5	3	0	0
5	4	0	0
5	5	1/36	25/36
5	6	2/36	60/36
6	1	0	0
6	2	0	0
6	3	0	0
6	4	0	0
6	5	0	0
6	6	1/36	36/36
Total			441/36

So,

---------------------

Following table shows the calculations for mean and var of U:

U	P(U=u)	u*P(U=u)	u^2*P(U=u)
1	1/36	1/36	1/36
2	3/36	6/36	12/36
3	5/36	15/36	45/36
4	7/36	28/36	112/36
5	9/36	45/36	225/36
6	11/36	66/36	396/36
Total		161/36	791/36

So,

--------------------------------------------------------

Following table shows the calculations for mean and var of V:

V	P(V=v)	v*P(V=v)	v^2*P(V=v)
1	11/36	11/36	11/36
2	9/36	18/36	36/36
3	7/36	21/36	63/36
4	5/36	20/36	80/36
5	3/36	15/36	75/36
6	1/36	6/36	36/36
Total		91/36	301/36

So,

The covariance is

Cov(U,V) = E(UV) - E(U) E(V) = 12.25 - (161/36) * (91/36) = 0.9452

Therefore correlation coefficient is

orchestra answered 2 years ago

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