urgent help need mean variance

[pasa]

Hi everybody,

If a random variable X is distributed in [0,1], can we say E[x]>var[x]?

If yes, can you please give me a reference which is stated that or can you please give me the proof?

I would be really thankful if you can help

 

[BGM]

Note that for any real number \( x \in (0, 1) \), and \( n > m \)

we have

\( x^n < x^m \)

Therefore we have

\( X^n < X^m \) for any random variables with support \( (0, 1) \)

and thus

\( E[X^n] < E[X^m] \)

In particular,

\( E[X] > E[X^2] > E[X^2] - E[X]^2 = Var[X] \)