Assume that \(F\) is the cumulative distribution function of a binomial random variable with parameters \(n\) and \(p\) equal to \(6\) and \(\frac{2}{3}\). Calculate \(F(3)\).

**(A)** \(\frac{26} { 27}\)
\(\quad\quad\) **(B)** \(\frac{ 233} {729} \)
\(\quad\quad\) **(C)** \(\frac{19}{27}\)
\(\quad\quad\) **(D)** \(\frac{19}{729}\)
\(\quad\quad\) **(E)** \(\frac{ 8 }{ 27}\)
\(\quad\quad\) **(F)** \(\frac{ 64 } {729 }\)
\(\quad\quad\) **(G)** \(\sqrt{2\pi}\)

**(A)**\(M=\int_{3}^{4}\int_{5+5(x-3)}^{5+6(x-3)}g(x,y)\,dydx+\int_{4}^{5}\int_{10+7(x-4)}^{5+6(x-3)}g(x,y)\,dydx\)**(B)**\(M=\int_{3}^{4}\int_{5+\frac{5}{2}(x-3)}^{5+12(x-3)}g(x,y)\,dydx + \int_{4}^{5}\int_{5+\frac{5}{2}(x-3)}^{17-7(x-4)}g(x,y)\,dydx\)**(C)**\(M=\int_{3}^{4}\int_{17-12(x-3)}^{17-\frac{7}2(x-3)}g(x,y)\,dydx+\int_{4}^{5}\int_{5+5(x-4)}^{17-\frac{7}2(x-3)}g(x,y)\,dydx\)**(D)**\(M=\int_{3}^{4}\int_{17-12(x-3)}^{17-7(x-3)}g(x,y)\,dxdy+\int_{4}^{5}\int_{10+7(x-4)}^{5+6(x-3)}g(x,y)\,dxdy\)**(E)**\(M=\int_{3}^{4}\int_{5+\frac{5}{2}(x-3)}^{5+12(x-3)}g(x,y)\,dxdy+\int_{4}^{5}\int_{10+7(x-4)}^{5+6(x-3)}g(x,y)\,dxdy\)**(F)**\(M=\int_{3}^{5}\int_{5 }^{17}g(x,y)\,dxdy \)**(G)**\(M=\int_{3}^{5}\int_{5 }^{17}g(x,y)\,dydx \)

**(A)**\(\int_{6}^{5\sqrt 2}\int_{72}^{2x^2}Cx^2y\,dydx + \int_{5\sqrt 2}^{1000}\int_{72}^{100}Cx^2y\,dydx\)**(B)**\(\int_{6}^{5\sqrt 2}\int_{72}^{2x^2}Cx^2y\,dydx + \int_{5\sqrt 2}^{1000}\int_{72}^{100}Cx^4y^2\,dydx\)**(C)**\(\int_{6\sqrt 2}^{10}\int_{72}^{x^2}Cx^2y\,dydx + \int_{10}^{1000}\int_{72}^{100}Cx^2y\,dydx\)**(D)**\(\int_{6}^{5\sqrt 2}\int_{72}^{2x^2}Cx^4y^2\,dydx + \int_{5\sqrt 2}^{1000}\int_{72}^{100}Cx^4y^2\,dydx\)**(E)**\(\int_{6\sqrt 2}^{10}\int_{72}^{x^2}Cx^4y^2\,dydx + \int_{10}^{1000}\int_{72}^{100}Cx^4y^2\,dydx\)