Easy if you know how...

[DavidHooper]

Choose any number between -1 and 1, with every number being equally likely.

Square this number.

What is the mean of this value?
How does everyone else work this kind of thing out - I would use a uniform distribution to calculate E(X^2).

[Guv]

Without doing a bit of research, I do not know what a uniform distribution is. Hence, I would not know how to apply it.

One plausible approach is to integrate x^2 from minus one to plus one and divide by the length of the interval. The integeral is x^3/3. The definite integral is 1/3 - (-1/3), which is 2/3. Dividing by two gives 1/3.

A reasonable answer is therefore 1/3, which may or may not be the only or the best answer. It is the best answer I can think of without doing some research.

If you consider the graph of x^2, the area between the curve and the X-Axis is equivalent to the area of a rectangle 2 units long and 1/3 unit high. In some sense, 1/3 is the average height of the figure.

[DavidHooper]

1/3 is correct - I wish i had your intuition!

The uniform distribution is just what it sounds like - every number has an equal probability of being chosen. The area under the graph must equal 1 as probs sum to 1. So the height is 1/2.

To find the mean (which is called E(X) by statisticians), then:

E(x)=Integral[x/2]
E(x^2)=Integral[X^2/2)
E(X^0.5)=Integral[X^0.5/2)

and the limits are 1 and -1.