Assume $U$~$\chi^2(5)$, $V$~$\chi^2(9)$, $Z$~$N(0,1)$, U, V, Z are mutually independent, calculate:
a. $P(Z > 0.611V^\frac{1}{2})$
b. $P(\frac{U}{V} < 1.933)$
c. Find a $c$ such that $P(\frac{U}{U+V} > c)=0.99$
d. Find a $d$ such that $P(\frac{U}{V} >d)=0.05$
- Since I written down:
- U: $f_5(u) = \frac{0.5^{2.5}}{\Gamma(2.5)}u^{1.5}e^{-0.5u}$, $x\geqq0$,
- V: $f_9(v) = \frac{0.5^{4.5}}{\Gamma(4.5)}v^{3.5}e^{-0.5v}$, $x\geqq0$,
- Z: $f(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$
Can I say $\frac{U}{V} = \frac{\Gamma(2)}{0.5^{2}}x^{-2}$ directly? and then calculate $\frac{\Gamma(2)}{0.5^{2}}v^{-2} = 0.99$ for example to find the c?