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Suppose you have an event that has 2-equally likely possibilities i.e, $X = a$ and $X = b$.

Clearly, $E(x)$ $=$ $(a+b)/2$ $=$ $1$

and $E(x^2)$ $=$ $(a^2 + b^2)/2$ $=$ $1$

Simplifying above two equations we get : $a + b = 2$ and $a^2 + b^2 = 2$

Putting $a = 2 - b$ on equation 2 and then solving $(2-b)^2 + b^2 = 2$, we get $a = 1$ and $b = 1$

Now,  E(X100) = (1100 + 1100)/2  = 1

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This question is based on probability distribution, you solved it like some apti question. E(x) is the mean.

E(X100) = E( (X50)2 )

Now, as we go on reducing, we get E(X25). Now, E(X25) = E( X24.X)

We know that E(A.B) = E(A). E(B) for independent events A and B.

So, E( X24.X) = E(X24). E(X) = 1    (I am not sure at this step)

reshown

E( X24.X) = E(X24). E(X) = 1

this is correct when one function is independent of other i.e E(x.y) =E(X).E(Y).(when x,y are independent)

CAN WE SAY X24 is independent of X ?

@Agrasar. I think I got the solution. Going with the definition of expectation,

if s={x1,x2.........,xn} is the sample space then,

E(X2) = $\sum_{r\epsilon s}^{}$ X(r=s)2.P(r)     ..................(1)

E(X) = $\sum_{r\epsilon s}^{}$ X(r=s).P(r)        ..................(2)

Now, (1) and (2) are both equal which implies X(r=s)2=X(r=s)       ..................(3)

Now, lets consider what value we get for E(X3).

E(X3)

= $\sum_{r\epsilon s}^{}$ X(r=s)3.P(r)

= $\sum_{r\epsilon s}^{}$ X(r=s)2. X(r=s) .P(r)

= $\sum_{r\epsilon s}^{}$ X(r=s). X(r=s) .P(r)     .................from (3)

= E(X2)

=1

Similarly, we can go on computing for E(X100).

Thus, I think, E(X100) = 1

it should be 1

E(x100) =E(x50 ) (becoz E(X50^2) =E(X100)

=E(X25)

=E(X12 . X)

=E(X6 . X)

=E(X3.X)

=E(X^4)

=E((x^2)^2)

=E(X^2)

=E(X)

=1
@cse23. They have give that E(X2)=E(X) and not E(X100)=E(X50). You are concluding that both things are same which is not correct I think unless they get proved by some formula.

So, both things are different. I had also done the same but later realised that its not correct.
but 100 =(50)^2 ryt?
@cse23. yes but there is difference between x and x^50