Given a random variable whose values are between $0$ and $1$, it is not difficult to see that the mean, or the expected value, is between $0$ and $1$, and the standard deviation is between $0$ and $\tfrac{1}{2}$.
However, not every combination is possible. If $\sigma=0$ then the mean could indeed be everywhere in $[0,1]$, but I tend to believe that $\sigma=\frac{1}{2}$ is only possible if $E[X] = \frac{1}{2}$.
Am I right? Are there simple restrictions involving a bounded variable's expected value and standard deviation?
Edit: Using the equality $\sigma(X) = \sigma(X-\frac{1}{2})$ we can get the equality $\sigma^2+(E[X]-\frac{1}{2})^2 = E[(X-\frac{1}{2})^2]$, the last expression being bounded by $0$ and $\frac{1}{4}$, so we get that the possible combinations must lie in the upper half circle (in the $\sigma-\mu$ plane) of radius $\frac{1}{2}$ around $\sigma=0,\mu=\frac{1}{2}$. Can we get a better result, or can any combination in the circle be reached?