Expected value and variance of the square root of a random variable Let $X$ be a univariate continuous random variable for which I can calculate all raw and central moments.
Is there an exact way to calculate $E[\,\sqrt{X}\,]$ and $\mathrm{Var}[\,\sqrt{X}\,]$ in this case?
That is to say, I am interested in calculating or estimating $E[\,g(X)\,]$ and $\mathrm{Var}[\,g(X)\,]$ when $g$ is the square root function.
I have asked this in a general way here: Approximating the expected value and variance of the function of a (continuous univariate) random variable . I have also read answers and coments to this question: Variance of powers of a random variable , but I think it refers to integer powers, which is not my case.
 A: If you only have the sequence of moments, the $\frac12$-th moment is not necessarily determined by them. 
If the MGF exists in a neighborhood of zero, then the moment sequence would determined the distribution and the $\frac12$-th moment should be determined (though not always amenable to algebraic calculation).
However if you have the pdf, we can avoid all that, since we can try to compute $E(X^\frac12)$ directly (e.g. by calculating the integral $\int_0^\infty x^\frac12 f(x) dx$ -- I presume $X$ is on the non-negative half line,for the obvious reason). Note also that the distribution of $\sqrt{X}$ will be very easy to  write down (if $Y=\sqrt{X},\, F_Y(y)=F_X(y^2)$ and $f_Y(y)=2yf_X(y^2)$). If, for example, we recognize that density as a standard one - it might be very fast to identify the expectation that way.
As whuber points out in comments, all we need to find is $E(\sqrt{X})$, since $\text{Var}(\sqrt{X})=E(X)-E(\sqrt{X})^2$.
A: if you are only interested in the upper bound of the expectation, you can use Jensen's Inequality to immediately upper bound $E[\sqrt{X}] \leq \sqrt{E[X]}$, if $E[X]$ is sufficiently close to 1, the approximation would be quite good.. 
Otherwise the standard tool to approximate the expectation is to use the Taylor series of $y = X-1$,  $\sqrt{1+y} = 1 - y/2 + y^2/8 - y^3/16 + 5y^4/128  \dots$
and since you already have the the MGF of $X$, you should be able to calculate that quickly...  
