1.2k views

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

1. There is a sample point at which $X$ has the value $5$.
2. There is a sample point at which $X$ has value greater than $5$.

3. There is a sample point at which $X$ has a value greater than equal to $5$.

4. None of the above

edited | 1.2k views
option elimination gives answer c .. bt i have one doubt if 'greater or equal to' is true then i think 'less than equal to' also true .. is it not  @Arjun sir  ??

Hello Digvijay

Yes! That's one possibility.

Possibility 1) one possibility is every variable is equal to 5 , hence average will be 5.

Possibility 2) if some variable is greater than 5 , then there must be some other variable that would be less than 5.

Possibility 3 ) Mixture of both above possibilities.

Expectation of discrete random variable (finite case)

$E(X) = x_1 p_1 + x_2p_2 + \dots +x_np_n$

$E(X) = 5, 0 \leq pi \leq1$

$p_1 + p_2 + \dots + p_n = 1$

Therefore, $E(X) = 5$ is possible only if at-least one of the  $x_i \geq 5$
selected by
in 1st eqn, xnpn rt?
@Arjun : thnx  ... corrected now ..
thats correct :)
:O how? You just gave an example case where a, b and c are true. But how can we say this for all cases?
c is true if and only if one of a or b is true. And either 'a' or 'b' is true, but not necessarily BOTH. i.e.,

c ↔ a ∨ b

but

c does not imply a and c does not imply b.
No. C is not the answer because it is stronger. A and B are wrong. C is the only answer.
Why cant the answer be There is a sample point at which x has value <=5 ?? Please correct me on this.
And if this answer is also acceptable then the answer must be d)None of these.

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 ...

edited
A gud view
I think ur statement "Expectation is nothing but average value " is nt correct every time ...
I think answer should be c) as I could not find any case in which mean is 5 and at all sample points X has value less than 5