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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.
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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  ??
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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.

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Mean, Median, Mode Calculator - Inch Calculator

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For this graph random variable X is greater than 5 . But if a graph is negatively skewed or symmetrical then value can be less than and equal to 5 respectively . So value can be greater than , less than  or equal to 5 .
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This is a Simple Diagram or generalized Diagarm

It has no relation with any Point.
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Below are the contradicting cases

(Bolded one is probability p and non-bolded is random variable x so written of form p*x) ---

E(x) =  0.8*6 + 0.2*1 = 5   (Option- A)

E(x) = 0.8*5  +  0.2*5  = 5 (Option – B)

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3 Answers

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31 votes
 
Best answer

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  $x_i \geq 5$

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5 Comments

But I have read that expectation means which outcome has highest chance to occur during experiment. But if every outcome is greater than 5 then how can the expectation be 5. Please explain accoridng to my concept
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First of all option B is not true. But you're interpreting it wrongly. I would suggest to do Mathematical logic first before doing any other subject.
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"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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53 votes
53 votes

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 ... 

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5 Comments

A gud view
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I think ur statement "Expectation is nothing but average value " is nt correct every time ...
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@puja why not? Expectation is nothing but mean or average.
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It's weighted average.
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yes
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3 votes
3 votes
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
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2 Comments

For example if X = {0,1,2,3,4}

and if P1=P2=P3=P4=P5 = ½

So E(X) = 0(1/2) + 1(1/2) + 2(1/2) + 3(1/2) + 4(1/2)  = 10(1/2) = 5. In this case none of the options are satisfied. So why is (d) not the correct answer ?
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Sum of probabilities p1, P2,....p5 should be equal to 1
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Answer:

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