A random variable X has a Beta$(a = 3.3, b = 4.2)$ distribution
(a) Compute $E(\sqrt{1-X})$
Answer:
(a)
$E(\sqrt{1-X}) = \frac{1}{0.01112334}\int_{0}^{1} \sqrt{1-x} x^{2.3} (1-x)^{3.2} dx = \int_{0}^{1} x^{2.3} (1-x)^{3.7} dx = \frac{1}{0.01112334} \frac{\Gamma{(3.3)} \Gamma{(4.7)}}{\Gamma{(8)}} = \frac{1}{0.0111233}\frac{(2.3)!(3.7)!}{7!}$
How do I solve for decimal factorials above?
Using the decimal value $0.01112334$ is a bad idea. Instead, use, $$E(\sqrt{1-X})=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_0^1x^{a-1}(1-x)^{1/2+b-1}dx=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\frac{\Gamma(a)\Gamma(b+1/2)}{\Gamma(a+b+1/2)}$$ that is, $$E(\sqrt{1-X})=\frac{\Gamma(7.5)\Gamma(4.7)}{\Gamma(4.2)\Gamma(8)}$$ This can be slightly simplified, using Legendre duplication formula $$\Gamma(x)\Gamma(x+1/2)=2^{1-2x}\sqrt\pi\,\Gamma(2x)$$ but not much (AFAICT, the most one can do is to reduce this to two unknown values of the $\Gamma$ function).