91影视

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

Given: \({\rm{\Gamma (6)}}\)

Definition gamma function: \({\rm{\Gamma (\alpha ) = }}\int_{\rm{0}}^{{\rm{ + \yen}}} {{{\rm{x}}^{{\rm{\alpha - 1}}}}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\)

Replace \({\rm{\alpha }}\) by\({\rm{6}}\): \({\rm{\Gamma (6) = }}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{x}}^{{\rm{6 - 1}}}}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx = }}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{x}}^{\rm{5}}}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\)

Use partial integral \(\int {\rm{u}} {\rm{dv = uv - }}\int {\rm{v}} {\rm{du}}\) with \({\rm{u = }}{{\rm{x}}^{\rm{5}}}{\rm{,du = 5}}{{\rm{x}}^{\rm{4}}}{\rm{,v = - }}{{\rm{e}}^{{\rm{ - x}}}}{\rm{,dv = }}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}_{{\rm{: }}}}}}\)

\(\begin{array}{l}{\rm{ = }}\left. {\left( {{\rm{ - }}{{\rm{x}}^{\rm{5}}}{{\rm{e}}^{{\rm{ - x}}}}} \right)} \right|_{\rm{0}}^{{\rm{ + \yen }}}{\rm{ - }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{ - }} {\rm{5}}{{\rm{x}}^{\rm{4}}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\\{\rm{ = 0 + }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{5}} {{\rm{x}}^{\rm{4}}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\\{\rm{ = 5}}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{x}}^{\rm{4}}}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\end{array}\)

Use partial integral \(\int {\rm{u}} {\rm{dv = uv - }}\int {\rm{v}} {\rm{du}}\) with\({\rm{u = }}{{\rm{x}}^{\rm{4}}}{\rm{,du = 4}}{{\rm{x}}^{\rm{3}}}{\rm{,v = - }}{{\rm{e}}^{{\rm{ - x}}}}{\rm{,dv = }}{{\rm{e}}^{{\rm{ - x}}}}\):

\(\begin{array}{l}{\rm{ = 5}}\left( {\left. {\left( {{\rm{ - }}{{\rm{x}}^{\rm{4}}}{{\rm{e}}^{{\rm{ - x}}}}} \right)} \right|_{\rm{0}}^{{\rm{ +\yen }}}{\rm{ - }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{ - }} {\rm{4}}{{\rm{x}}^{\rm{3}}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 5}}\left( {{\rm{0 + }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{4}} {{\rm{x}}^{\rm{3}}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 20}}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{x}}^{\rm{3}}}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\end{array}\)

Use partial integral \(\int {\rm{u}} {\rm{dv = uv - }}\int {\rm{v}} {\rm{du}}\) with \({\rm{u = }}{{\rm{x}}^{\rm{3}}}{\rm{,du = 3}}{{\rm{x}}^{\rm{2}}}{\rm{,v = - }}{{\rm{e}}^{{\rm{ - x}}}}{\rm{,dv = }}{{\rm{e}}^{{\rm{ - x}}}}\) :

\(\begin{array}{l}{\rm{ = 20}}\left( {\left. {\left( {{\rm{ - }}{{\rm{x}}^{\rm{3}}}{{\rm{e}}^{{\rm{ - x}}}}} \right)} \right|_{\rm{0}}^{{\rm{ + \yen }}}{\rm{ - }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{ - }} {\rm{3}}{{\rm{x}}^{\rm{2}}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 20}}\left( {{\rm{0 + }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{3}} {{\rm{x}}^{\rm{2}}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 60}}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{x}}^{\rm{2}}}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\end{array}\)

Use partial integral \(\int {\rm{u}} {\rm{dv = uv - }}\int {\rm{v}} {\rm{du}}\) with \({\rm{u = }}{{\rm{x}}^{\rm{2}}}{\rm{,du = 2x,v = - }}{{\rm{e}}^{{\rm{ - x}}}}{\rm{,dv = }}{{\rm{e}}^{{\rm{ - x}}}}\) :

\(\begin{array}{l}{\rm{ = 60}}\left( {\left. {\left( {{\rm{ - }}{{\rm{x}}^{\rm{2}}}{{\rm{e}}^{{\rm{ - x}}}}} \right)} \right|_{\rm{0}}^{{\rm{ + \yen }}}{\rm{ - }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{ - }} {\rm{2x}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 60}}\left( {{\rm{0 + }}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{2}} {\rm{x}}{{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 120}}\int_{\rm{0}}^{{\rm{ + \yen }}} {\rm{x}} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}\end{array}\)

Use partial integral \(\int {\rm{u}} {\rm{dv = uv - }}\int {\rm{v}} {\rm{du}}\) with \({\rm{u = x,du = 1,v = - }}{{\rm{e}}^{{\rm{ - x}}}}{\rm{,dv = }}{{\rm{e}}^{{\rm{ - x}}}}\) :

\(\begin{array}{l}{\rm{ = 120}}\left( {\left. {\left( {{\rm{ - x}}{{\rm{e}}^{{\rm{ - x}}}}} \right)} \right|_{\rm{0}}^{{\rm{ + \yen }}}{\rm{ - }}\int_{\rm{0}}^{{\rm{ + \yen}}} {\rm{ - }} {{\rm{e}}^{{\rm{ - x}}}}{\rm{dx}}} \right)\\{\rm{ = 120}}\left( {{\rm{0 + }}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{e}}^{{\rm{ - x}}}}} {\rm{dx}}} \right)\\{\rm{ = 120}}\int_{\rm{0}}^{{\rm{ + \yen }}} {{{\rm{e}}^{{\rm{ - x}}}}} {\rm{dx}}\\{\rm{ = }}\left. {{\rm{120}}\left( {{\rm{ - }}{{\rm{e}}^{{\rm{ - x}}}}} \right)} \right|_{\rm{0}}^{{\rm{ + \yen }}}\\{\rm{ = 120(1)}}\\{\rm{ = 120}}\end{array}\)

Thus, we have obtained\({\rm{\Gamma (6) = 120}}\).

ALTERNATIVE SOLUTION

Property gamma function for \({\rm{n}}\) integer: \({\rm{\Gamma (n) = (n - 1)!}}\)

Replace \({\rm{n}}\) by 6: \({\rm{\Gamma (6) = (6 - 1)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120}}\)

b)

Given: \({\rm{\Gamma }}\left( {\frac{{\rm{5}}}{{\rm{2}}}} \right)\)

Properties gamma function \({\rm{(\alpha > 1)}}\) : \({\rm{\Gamma (\alpha ) = (\alpha - 1), \Gamma (\alpha - 1)\Gamma }}\left( {\frac{{\rm{1}}}{{\rm{2}}}} \right)\)

Using these properties, we then obtain for \({\rm{\alpha = }}\frac{{\rm{5}}}{{\rm{2}}}{\rm{ = }}\)

\(\begin{array}{c}{\rm{\Gamma }}\left( {\frac{{\rm{5}}}{{\rm{2}}}} \right){\rm{ = }}\left( {\frac{{\rm{5}}}{{\rm{2}}}{\rm{ - 1}}} \right)\\{\rm{\Gamma }}\left( {\frac{{\rm{5}}}{{\rm{2}}}{\rm{ - 1}}} \right){\rm{ = }}\frac{{\rm{3}}}{{\rm{2}}}\\{\rm{\Gamma }}\left( {\frac{{\rm{3}}}{{\rm{2}}}} \right){\rm{ = }}\frac{{\rm{3}}}{{\rm{2}}}\left( {\frac{{\rm{3}}}{{\rm{2}}}{\rm{ - 1}}} \right)\\{\rm{\Gamma }}\left( {\frac{{\rm{3}}}{{\rm{2}}}{\rm{ - 1}}} \right){\rm{ = }}\frac{{\rm{3}}}{{\rm{2}}}\frac{{\rm{1}}}{{\rm{2}}}\\{\rm{\Gamma }}\left( {\frac{{\rm{1}}}{{\rm{2}}}} \right){\rm{ = }}\frac{{\rm{3}}}{{\rm{2}}}\frac{{\rm{1}}}{{\rm{2}}}\sqrt {\rm{\pi }} \\{\rm{ = }}\frac{{{\rm{3}}\sqrt {\rm{\pi }} }}{{\rm{4}}}\end{array}\)

c)

Given: \({\rm{F}}\)represents the incomplete gamma function

\(\begin{array}{l}{\rm{F(4;5)}}\\{\rm{F(5;4)}}\end{array}\)

\({\rm{F(4;5)}}\)is the incomplete gamma function with \({\rm{x = 4}}\) and\({\rm{\alpha = 5}}\). The value can be found in the incomplete gamma function table in the appendix (table \({\rm{\;A}}{\rm{. 4}}\) in my book) in the row with \({\rm{x = 4}}\) and in the column with\({\rm{\alpha = 5}}\):

\({\rm{F(4;5) = 0}}{\rm{.371}}\)

\({\rm{F(5;4)}}\)is the incomplete gamma function with \({\rm{x = 5}}\) and\({\rm{\alpha = 4}}\). The value can be found in the incomplete gamma function table in the appendix (table A.4 in my book) in the row with \({\rm{x = 5}}\) and in the column with \({\rm{\alpha = 4}}\) :

\({\rm{F(5;4) = 0}}{\rm{.735}}\)

d)

Given: \({\rm{X}}\)has a standard gamma distribution

\(\begin{aligned}{l}{\rm{\alpha = 7}} P(X拢5) \end{aligned}\)

Property gamma distribution:

\(P(X拢 x) = F(x;\alpha ,\beta ) = F\left( {\frac{{\rm{x}}}{{\rm{\beta }}}{\rm{,\alpha }}} \right)\)

\({\rm{F}}\)represents the incomplete gamma function.

Use this property for\({\rm{x = 5}}\). The standard gamma distribution has\({\rm{\beta = 1}}\).

\(P(X拢 5) = F \left( {\frac{{\rm{5}}}{{\rm{1}}}{\rm{,7}}} \right){\rm{ = F(5,7)}}\)

\({\rm{F(5;7)}}\)is the incomplete gamma function with \({\rm{x = 5}}\) and\({\rm{\alpha = 7}}\). The value can be found in the incomplete gamma function table in the appendix (table \({\rm{A}}{\rm{.4}}\)) in the row with \({\rm{x = 5}}\) and in the column with \({\rm{\alpha = 7}}\):

\({\rm{F(5;7) = 0}}{\rm{.238}}\)

e)

Given: \({\rm{X}}\)has a standard gamma distribution

\(\begin{array}{l}{\rm{\alpha = 7}}\\{\rm{P(3 < X < 8)}}\end{array}\)

Property gamma distribution:

\({\rm{P(X拢 x) = F(x;\alpha ,\beta ) = F}}\left( {\frac{{\rm{x}}}{{\rm{\beta }}}{\rm{,\alpha }}} \right)\)

\({\rm{F}}\)represents the incomplete gamma function.

Use this property for \({\rm{x = 3}}\) and\({\rm{x = 8}}\). The standard gamma distribution has\({\rm{\beta = 1}}\).

\(\begin{array}{l}{\rm{P(X拢 3) = F}}\left( {\frac{{\rm{3}}}{{\rm{1}}}{\rm{,7}}} \right){\rm{ = F(3,7)}}\\{\rm{P(X拢 8) = F}}\left( {\frac{{\rm{8}}}{{\rm{1}}}{\rm{,7}}} \right){\rm{ = F(8,7)}}\end{array}\)

\({\rm{F(3;7)}}\)is the incomplete gamma function with \({\rm{x = 3}}\) and \({\rm{\alpha = 7}}\). The value can be found in the incomplete gamma function table in the appendix (table \({\rm{A}}{\rm{.4}}\)) in the row with \({\rm{x = 3}}\) and in the column with \({\rm{\alpha = 7}}\) :

\({\rm{F(3;7) = 0}}{\rm{.034}}\)

\({\rm{F(8;7)}}\)is the incomplete gamma function with \({\rm{x = 8}}\) and \({\rm{\alpha = 7}}\). The value can be found in the incomplete gamma function table in the appendix (table \({\rm{A}}{\rm{.4}}\)) in the row with \({\rm{x = 8}}\) and in the column with \({\rm{\alpha = 7}}\):

\({\rm{F(8;7) = 0}}{\rm{.687}}\)

The probability between two boundaries is the difference between the probabilities to the left of the boundaries:

\(\begin{array}{c}{\rm{P(3 < X < 8) = P(X < 8) - P(X拢 3)}}\\{\rm{ = P(X拢 8) - P(X拢 3)}}\\{\rm{ = 0}}{\rm{.687 - 0}}{\rm{.034}}\\{\rm{ = 0}}{\rm{.653}}\\{\rm{ = 65}}{\rm{.3\% }}\end{array}\)