GATE | GATE CS 1999 | Question 2

Suppose that the expectation of a random variable X is 5. Which of the following statement is True?

 
(A) There is a simple point at which X has the value 5.
(B) There is a simple point at which X has value greater than 5.
(C) There is a simple point at which X has a value greater than or equal to 5.
(D) None of the above.


Answer: (C)

Explanation: E(X)=\int \limits_{-\infty}^{\infty} xf(x)\:dx
The first option is wrong since the Expected value need not be present in the data.
The second option is wrong since in the event where every value in the data is the same, the expected value is the same as every value. So, In this case, there is no value which is greater than the expected value.
The third option is correct, Since there must be some value which is either equal to or greater than the expected value. This is because it is impossible to have all the values less than the expected value as it is analogous to the mean. And the mean value is never greater than all the values of the data.

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