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- If the area of a square is uniformly distributed over an interval, why is the expected value of the length different to the square root of the expected value of the square's area?

Hello,

I am currently stumped over a question that has to do with the continuous uniform distribution. The question was taken from a stats exam, and while I understand the solution given in the mark scheme, I don't understand why my way of thinking doesn't work.

The problem is:

The sides of a square are of length L cm and its area is A cm^2. Given that A is uniformly distributed on the interval [15, 20], find E(L).

The mark scheme solution uses integration [ by writing L = A^(1/2) ] so that E(L) = ∫(a^(1/2)) x (1/5)da between 15 and 20. I appreciate that this is making use of the fact that E[g(x)] = ∫g(x)f(x) dx for a CRV.

On the other hand, why can't we simply find E(A) and take its square root? If we expect the area to be E(A), is the length when it has this area not equal to the expected length?

If somebody could clarify this I would be very grateful for your help.

PS it just so happens in this case that both methods give you the same answer, but I know that my way is wrong and the mark scheme won't allow it.

I am currently stumped over a question that has to do with the continuous uniform distribution. The question was taken from a stats exam, and while I understand the solution given in the mark scheme, I don't understand why my way of thinking doesn't work.

The problem is:

The sides of a square are of length L cm and its area is A cm^2. Given that A is uniformly distributed on the interval [15, 20], find E(L).

The mark scheme solution uses integration [ by writing L = A^(1/2) ] so that E(L) = ∫(a^(1/2)) x (1/5)da between 15 and 20. I appreciate that this is making use of the fact that E[g(x)] = ∫g(x)f(x) dx for a CRV.

On the other hand, why can't we simply find E(A) and take its square root? If we expect the area to be E(A), is the length when it has this area not equal to the expected length?

If somebody could clarify this I would be very grateful for your help.

PS it just so happens in this case that both methods give you the same answer, but I know that my way is wrong and the mark scheme won't allow it.