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Question
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denotes the larger of the two numbers obtained. Find E(X).
Solution
The two positive integers can be selected from the first six positive integers without replacement in 6 × 5 = 30 ways
X represents the larger of the two numbers obtained. Therefore, X can take the value of 2, 3, 4, 5, or 6.
For X = 2, the possible observations are (1, 2) and (2, 1).
:. `P(X=2) = 2/30 = 1/15`
For X = 3, the possible observations are (1, 3), (2, 3), (3, 1), and (3, 2).
:. `P(X=3) = 4/30 = 2/15`
For X = 4, the possible observations are (1, 4), (2, 4), (3, 4), (4, 3), (4, 2), and (4, 1).
:. `P(X=4) = 6/30 = 1/5`
For X = 5, the possible observations are (1, 5), (2, 5), (3, 5), (4, 5), (5, 4), (5, 3), (5, 2), and (5, 1).
:. `P(X=5) = 8/30 = 4/15`
For X = 6, the possible observations are (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6,5) , (6, 4), (6, 3), (6, 2), and (6, 1).
:. `P(X=6) = 10/30 = 1/3`
Therefore, the required probability distribution is as follows.
| X | 2 | 3 | 4 | 5 | 6 |
| P(X) | `1/15` | `2/15` | `1/5` | `4/15` | `1/3` |
Then, E(X) = Σ xi P(xi)
= `2xx1/15+3xx2/15+4xx3/15+5xx4/15+6xx5/15`
= `(2+6+12+20+30)/15`
= `70/15`
=`14/3`
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