Suppose that the expectation of a random variable X is 5. Which of the…
1999
Suppose that the expectation of a random variable X is 5. Which of the following statements is true?
- A.
There is a sample point at which X has the value 5.
- B.
There is a sample point at which X has a value greater than 5.
- C.
There is a sample point at which X has a value greater than or equal to 5.
- D.
None of the above.
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Correct answer: C
The correct answer is: There is a sample point at which X has a value greater than or equal to 5.
The expectation E[X] is a weighted average of the values taken by X. If X were less than 5 at every sample point, then its weighted average would also be less than 5, contradicting E[X] = 5.
So at least one sample point must have X ≥ 5.
However, X need not actually take the value 5. For example, X may take values 4 and 6 with equal probability, giving E[X] = 5, but X is never equal to 5.
Also, X need not be greater than 5 anywhere. For example, if X is always 5, then E[X] = 5 but no value is greater than 5.