urgent help need mean variance

pasa

New Member
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

TS Contributor
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,

$$E[X] > E[X^2] > E[X^2] - E[X]^2 = Var[X]$$