Suppose we roll a fair dice thrice in such a way that the assumption of random...

Suppose we roll a fair dice thrice in such a way that the assumption of random sampling is satisfied. Let {X1;X2;X3} be the numbers coming up in the first three throws.

Let their sample average be X3= (1/3) * ((X1+X2+X3))

1) Write down the sampling distribution ofX3. To this end, you need to write down a table that contains all values that X3 can take on together with their probabilities.

2) What is the probability that X3 greater than equal to 5, given that the dice is fair?

Expert Solution

1.

X1	X2	X3
1	1	1	1
1	1	2	1.33
1	1	3	1.67
1	1	4	2
1	1	5	2.33
1	1	6	2.67
1	2	1	1.33
1	2	2	1.67
1	2	3	2
1	2	4	2.33
1	2	5	2.67
1	2	6	3
1	3	1	1.67
1	3	2	2
1	3	3	2.33
1	3	4	2.67
1	3	5	3
1	3	6	3.33
1	4	1	2
1	4	2	2.33
1	4	3	2.67
1	4	4	3
1	4	5	3.33
1	4	6	3.67
1	5	1	2.33
1	5	2	2.67
1	5	3	3
1	5	4	3.33
1	5	5	3.67
1	5	6	4
1	6	1	2.67
1	6	2	3
1	6	3	3.33
1	6	4	3.67
1	6	5	4
1	6	6	4.33
2	1	1	1.33
2	1	2	1.67
2	1	3	2
2	1	4	2.33
2	1	5	2.67
2	1	6	3
2	2	1	1.67
2	2	2	2
2	2	3	2.33
2	2	4	2.67
2	2	5	3
2	2	6	3.33
2	3	1	2
2	3	2	2.33
2	3	3	2.67
2	3	4	3
2	3	5	3.33
2	3	6	3.67
2	4	1	2.33
2	4	2	2.67
2	4	3	3
2	4	4	3.33
2	4	5	3.67
2	4	6	4
2	5	1	2.67
2	5	2	3
2	5	3	3.33
2	5	4	3.67
2	5	5	4
2	5	6	4.33
2	6	1	3
2	6	2	3.33
2	6	3	3.67
2	6	4	4
2	6	5	4.33
2	6	6	4.67
3	1	1	1.67
3	1	2	2
3	1	3	2.33
3	1	4	2.67
3	1	5	3
3	1	6	3.33
3	2	1	2
3	2	2	2.33
3	2	3	2.67
3	2	4	3
3	2	5	3.33
3	2	6	3.67
3	3	1	2.33
3	3	2	2.67
3	3	3	3
3	3	4	3.33
3	3	5	3.67
3	3	6	4
3	4	1	2.67
3	4	2	3
3	4	3	3.33
3	4	4	3.67
3	4	5	4
3	4	6	4.33
3	5	1	3
3	5	2	3.33
3	5	3	3.67
3	5	4	4
3	5	5	4.33
3	5	6	4.67
3	6	1	3.33
3	6	2	3.67
3	6	3	4
3	6	4	4.33
3	6	5	4.67
3	6	6	5
4	1	1	2
4	1	2	2.33
4	1	3	2.67
4	1	4	3
4	1	5	3.33
4	1	6	3.67
4	2	1	2.33
4	2	2	2.67
4	2	3	3
4	2	4	3.33
4	2	5	3.67
4	2	6	4
4	3	1	2.67
4	3	2	3
4	3	3	3.33
4	3	4	3.67
4	3	5	4
4	3	6	4.33
4	4	1	3
4	4	2	3.33
4	4	3	3.67
4	4	4	4
4	4	5	4.33
4	4	6	4.67
4	5	1	3.33
4	5	2	3.67
4	5	3	4
4	5	4	4.33
4	5	5	4.67
4	5	6	5
4	6	1	3.67
4	6	2	4
4	6	3	4.33
4	6	4	4.67
4	6	5	5
4	6	6	5.33
5	1	1	2.33
5	1	2	2.67
5	1	3	3
5	1	4	3.33
5	1	5	3.67
5	1	6	4
5	2	1	2.67
5	2	2	3
5	2	3	3.33
5	2	4	3.67
5	2	5	4
5	2	6	4.33
5	3	1	3
5	3	2	3.33
5	3	3	3.67
5	3	4	4
5	3	5	4.33
5	3	6	4.67
5	4	1	3.33
5	4	2	3.67
5	4	3	4
5	4	4	4.33
5	4	5	4.67
5	4	6	5
5	5	1	3.67
5	5	2	4
5	5	3	4.33
5	5	4	4.67
5	5	5	5
5	5	6	5.33
5	6	1	4
5	6	2	4.33
5	6	3	4.67
5	6	4	5
5	6	5	5.33
5	6	6	5.67
6	1	1	2.67
6	1	2	3
6	1	3	3.33
6	1	4	3.67
6	1	5	4
6	1	6	4.33
6	2	1	3
6	2	2	3.33
6	2	3	3.67
6	2	4	4
6	2	5	4.33
6	2	6	4.67
6	3	1	3.33
6	3	2	3.67
6	3	3	4
6	3	4	4.33
6	3	5	4.67
6	3	6	5
6	4	1	3.67
6	4	2	4
6	4	3	4.33
6	4	4	4.67
6	4	5	5
6	4	6	5.33
6	5	1	4
6	5	2	4.33
6	5	3	4.67
6	5	4	5
6	5	5	5.33
6	5	6	5.67
6	6	1	4.33
6	6	2	4.67
6	6	3	5
6	6	4	5.33
6	6	5	5.67
6	6	6	6

	Frequency=f
1	1	0.0046
1.33	3	0.0139
1.67	6	0.0278
2	10	0.0463
2.33	15	0.0694
2.67	21	0.0972
3	25	0.1157
3.33	27	0.125
3.67	27	0.125
4	25	0.1157
4.33	21	0.0972
4.67	15	0.0694
5	10	0.0463
5.33	6	0.0278
5.67	3	0.0139
6	1	0.0046

2.

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