GATE CSE 1999 | Question: 1.1

Question

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

Comments

for this question answer can be all of the 3 options since they didnot mentioned always

if they specifically mention in all the cases which is must be true then option could be c

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for this question answer can be all of the 3 options since they didnot mentioned always

if they specifically mention in all the cases which is must be true then option could be c

i hope you got it

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if they ask which of the options could be true then A, B, C can be true but they asked which are true so only option C is true. @iiscwallah

Answer 1

Answer is (C)

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  $x_i \geq 5$

Comments

"has a value greater than equal to 5" does not say that all values are greater than 5.

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Thanks. Now I got it

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1. Expectation as an average:
   The expectation E[X]=5 is the weighted average of all possible values of X. For the average to be 5:

   • Some values of X must be ≥5, and some must be ≤5, unless all values are exactly 5.

   • If all sample points had X<5, then E[X]<5, which contradicts E[X]=5.

Answer 2

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 ... 

Comments

It's weighted average.

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yes

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

Answer 3

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

Comments

Sum of probabilities p1, P2,....p5 should be equal to 1

Answer 4

Statement of one of lecture of sachir sir credit @GO Classes 
expectation is avegrage while considering large enough sample space.

 

Answer 5

When the average is 5. Then there exists some sample points all around the value of 5 !

There should be some sample points <=5, some >=5, some < 5, some > 5.
There may or may not be any sample points = 5 as mean is a theoretical value.

 

I think the answer should be (D).

 

@Arjun, sir, people from the community, have I gone wrong somewhere ?

Comments

As correctly mentioned by others in the above comment, Average and Expectation are not the same.
 

Average is calculated on sample data.
Expectation is weighted average.

This might be a good read:
https://math.stackexchange.com/a/1355657

Answer 6

Answer is (C).