Let X be a random variable with V ( X ) = 25It also satisfies P ( X &gt; 20 ) &gt; 1 4 Which of the following is true?- E ( X ) &gt; 10- 5 &lt; E ( X ) ≤ 10- 0 &lt; E ( X ) ≤ 5- E ( X ) ≤ 0I tried using Chebyshev's inequality and found out that E ( X ) &gt; 5 but wasn't sure how to go on about it.

Question

Let X be a random variable with       V    (  X  )  =  25It also satisfies       P    (  X  &amp;gt;  20  )  &amp;gt;      1    4  Which of the following is true?-       E    (  X  )  &amp;gt;  10-   5  &amp;lt;      E    (  X  )  ≤  10-   0  &amp;lt;      E    (  X  )  ≤  5-       E    (  X  )  ≤  0I tried using Chebyshev&#039;s inequality and found out that       E    (  X  )  &amp;gt;  5 but wasn&#039;t sure how to go on about it.

occuffick24 · Accepted Answer

Step 1As a fairly major hint, you should use Chebyshev&#039;s inequality to show      E    X  ≥  c  −                              V                X                              P                (        X        ≥        c        )            Step 2(provided       P    (  X  ≥  c  )  ≠  0)

ajanlr · Accepted Answer

Step 1There cannot be an upper limit on E(X) - for example there is nothing to stop the distribution being centred around 100 or 1000. So option A is the only possibility.Step 2We can check option A by trying to make E(X) as small as possible. Put   p  (  X  =  20  )  =  0.25 and   p  (  X  =  a  )  =  0.75. Then  E  (  X  )  =  0.25  ∗  20  +  0.75  ∗  a  =  5  +  0.75  a  E  (      X    2    )  =  0.25  ∗      20    2    +  0.75  ∗      a    2    =  100  +  0.75      a    2    V  a  r  (  X  )  =  100  +  0.75      a    2    −  (  5  +  0.75  a      )    2    =  25which gives   a  ≈  8.45 and   E  (  X  )  ≈  11.3. So option A is satisfied.

Let X be a random variable with V(X)=25

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