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Suppose that the expectation of a random variable $X$ is $5$. Which of the following statements is true?

- There is a sample point at which $X$ has the value $5$.
- There is a sample point at which $X$ has value greater than $5$.
- There is a sample point at which $X$ has a value greater than equal to $5$.
- None of the above.

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Expectation is nothing but average value ...

A) Let the samples be** 4,6** .. Average = (4+6) / 2 **= 5** .. So average of 5 is possible even if no sample point has a value 5... **So A) is eliminated.**

B) Let samples be **5,5** .. Average = (5+5) / 2** = 5** .. So Average can be 5 even if no sample is greater than 5 .. **So B) is eliminated ..**.

C) should be always TRUE .. If you dont have even 1 sample which is** <= average_value x** then average of x is not possible.. This is also applicable if you dont have even 1 sample point which is** >= average_value**

So **Option C)** is the answer ...

@Aakash_ mean and average are calculated on sample data . but expectation is calculated on real probability . Its a theoretical concept.

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