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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
edited | 2.2k views
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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.

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  $x_i \geq 5$

edited
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in 1st eqn, xnpn rt?
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@Arjun : thnx  ... corrected now ..
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thats correct :)
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so , answer should be (a) , (b) and (c) are TRUE .
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:O how? You just gave an example case where a, b and c are true. But how can we say this for all cases?
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option (c) contains either option (a) or (b) ... so , ??
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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.
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for exactly one point exactly two options will be true , but option c more stronger  ,..?
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No. C is not the answer because it is stronger. A and B are wrong. C is the only answer.
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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.
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Sir, how option B is true? Please explain it I am not getting that option. Will option B not make expectation value greater than 5?
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Option B : There is a sample point at which X  has value greater than 5 .

suppose consider n= 3 and $x_1=x_2=x_3=5$

E(x) = $x_1p_1$  + $x_2p_2$ + $x_3p_3$  = 5

Hence,  E(X) = 5 is also possible when none of the sample points has value greater than 5.  Option(B) is false.
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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

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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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.
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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