$alpha$-Quantile for Gamma distribution $Gamma(n,frac{1}{2})$












1












$begingroup$


Let $XsimGamma(n,frac{1}{2})$ and $alphain(0,1)$. I know that



begin{align}
P(Xle s)&=int_0^ sfrac{1}{2^nGamma(n)}x^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{2(n-1)!}int_0^sBig(frac{x}{2}Big)^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{(n-1)!}int_0^{s/2} t^{n-1} e^{-t}dt\
end{align}



Now I wonder how can I find $u,v$ such that



$$frac{alpha}{2}ge frac{1}{(n-1)!}int_0^{u/2} t^{n-1} e^{-t}dt=P(Xle u),$$
$$1-frac{alpha}{2} le frac{1}{(n-1)!}int_0^{v/2} t^{n-1} e^{-t}dt= P(Xle v)text{ ?}$$
Maybe there is an easy argument... Thanks in advance for any help!






real-analysis probability analysis probability-theory statistics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Of course, the integral in the end is the same as in the definition of the Gammafunction.
    $endgroup$
    – user408858
    Dec 18 '18 at 0:12










  • $begingroup$
    If I understand your notation and if you need the answers instead of hints how to do the calculus, then here is R code that will give you the answer for $n = 1, 2, dots, 50$ and $alpha = 0.05:$ n = 1:50; L = qgamma(.025, n, 2); U = qgamma(.975, n ,2); cbind(n, L, U). For $n = 1,$ you have the exponential distribution, for which the calculus is trivial. For $n > 200,$ you should get good answers with normal approximation.
    $endgroup$
    – BruceET
    Dec 18 '18 at 0:28












  • $begingroup$
    Oh! Do I get you right, that there are these tables in backmatter of statistic books for exactly these kind of questions? :-)
    $endgroup$
    – user408858
    Dec 18 '18 at 0:30












  • $begingroup$
    You might be able to use widely available printed tables of chi-squared distributions (with appriat modifications; halve DF to get gamma shape). Some printed tables have only the upper tail. Look around. If you're allergic to R, I suppose MatLab and such will do much the same computations. // But R is free, devotedly bug free for basic stuff like this, and trivial to install. See Wikipedia. // Code qchisq(.95, 4) and qgamma(.95, 2, 1/2) both return 9.487729
    $endgroup$
    – BruceET
    Dec 18 '18 at 1:08












  • $begingroup$
    Actually one can just take the $1-frac{alpha}{2}$-quantile by definition(?) or just calculate it like @BruceET did. :-)
    $endgroup$
    – user408858
    Dec 20 '18 at 0:46
















1












$begingroup$


Let $XsimGamma(n,frac{1}{2})$ and $alphain(0,1)$. I know that



begin{align}
P(Xle s)&=int_0^ sfrac{1}{2^nGamma(n)}x^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{2(n-1)!}int_0^sBig(frac{x}{2}Big)^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{(n-1)!}int_0^{s/2} t^{n-1} e^{-t}dt\
end{align}



Now I wonder how can I find $u,v$ such that



$$frac{alpha}{2}ge frac{1}{(n-1)!}int_0^{u/2} t^{n-1} e^{-t}dt=P(Xle u),$$
$$1-frac{alpha}{2} le frac{1}{(n-1)!}int_0^{v/2} t^{n-1} e^{-t}dt= P(Xle v)text{ ?}$$
Maybe there is an easy argument... Thanks in advance for any help!






real-analysis probability analysis probability-theory statistics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Of course, the integral in the end is the same as in the definition of the Gammafunction.
    $endgroup$
    – user408858
    Dec 18 '18 at 0:12










  • $begingroup$
    If I understand your notation and if you need the answers instead of hints how to do the calculus, then here is R code that will give you the answer for $n = 1, 2, dots, 50$ and $alpha = 0.05:$ n = 1:50; L = qgamma(.025, n, 2); U = qgamma(.975, n ,2); cbind(n, L, U). For $n = 1,$ you have the exponential distribution, for which the calculus is trivial. For $n > 200,$ you should get good answers with normal approximation.
    $endgroup$
    – BruceET
    Dec 18 '18 at 0:28












  • $begingroup$
    Oh! Do I get you right, that there are these tables in backmatter of statistic books for exactly these kind of questions? :-)
    $endgroup$
    – user408858
    Dec 18 '18 at 0:30












  • $begingroup$
    You might be able to use widely available printed tables of chi-squared distributions (with appriat modifications; halve DF to get gamma shape). Some printed tables have only the upper tail. Look around. If you're allergic to R, I suppose MatLab and such will do much the same computations. // But R is free, devotedly bug free for basic stuff like this, and trivial to install. See Wikipedia. // Code qchisq(.95, 4) and qgamma(.95, 2, 1/2) both return 9.487729
    $endgroup$
    – BruceET
    Dec 18 '18 at 1:08












  • $begingroup$
    Actually one can just take the $1-frac{alpha}{2}$-quantile by definition(?) or just calculate it like @BruceET did. :-)
    $endgroup$
    – user408858
    Dec 20 '18 at 0:46














1












1








1





$begingroup$


Let $XsimGamma(n,frac{1}{2})$ and $alphain(0,1)$. I know that



begin{align}
P(Xle s)&=int_0^ sfrac{1}{2^nGamma(n)}x^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{2(n-1)!}int_0^sBig(frac{x}{2}Big)^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{(n-1)!}int_0^{s/2} t^{n-1} e^{-t}dt\
end{align}



Now I wonder how can I find $u,v$ such that



$$frac{alpha}{2}ge frac{1}{(n-1)!}int_0^{u/2} t^{n-1} e^{-t}dt=P(Xle u),$$
$$1-frac{alpha}{2} le frac{1}{(n-1)!}int_0^{v/2} t^{n-1} e^{-t}dt= P(Xle v)text{ ?}$$
Maybe there is an easy argument... Thanks in advance for any help!






real-analysis probability analysis probability-theory statistics







share|cite|improve this question











$endgroup$




Let $XsimGamma(n,frac{1}{2})$ and $alphain(0,1)$. I know that



begin{align}
P(Xle s)&=int_0^ sfrac{1}{2^nGamma(n)}x^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{2(n-1)!}int_0^sBig(frac{x}{2}Big)^{n-1}e^{-frac{x}{2}}dx\
&=frac{1}{(n-1)!}int_0^{s/2} t^{n-1} e^{-t}dt\
end{align}



Now I wonder how can I find $u,v$ such that



$$frac{alpha}{2}ge frac{1}{(n-1)!}int_0^{u/2} t^{n-1} e^{-t}dt=P(Xle u),$$
$$1-frac{alpha}{2} le frac{1}{(n-1)!}int_0^{v/2} t^{n-1} e^{-t}dt= P(Xle v)text{ ?}$$
Maybe there is an easy argument... Thanks in advance for any help!






real-analysis probability analysis probability-theory statistics




real-analysis probability analysis probability-theory statistics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 18 '18 at 0:21







user408858

















asked Dec 18 '18 at 0:11









user408858user408858

482213




482213












  • $begingroup$
    Of course, the integral in the end is the same as in the definition of the Gammafunction.
    $endgroup$
    – user408858
    Dec 18 '18 at 0:12










  • $begingroup$
    If I understand your notation and if you need the answers instead of hints how to do the calculus, then here is R code that will give you the answer for $n = 1, 2, dots, 50$ and $alpha = 0.05:$ n = 1:50; L = qgamma(.025, n, 2); U = qgamma(.975, n ,2); cbind(n, L, U). For $n = 1,$ you have the exponential distribution, for which the calculus is trivial. For $n > 200,$ you should get good answers with normal approximation.
    $endgroup$
    – BruceET
    Dec 18 '18 at 0:28












  • $begingroup$
    Oh! Do I get you right, that there are these tables in backmatter of statistic books for exactly these kind of questions? :-)
    $endgroup$
    – user408858
    Dec 18 '18 at 0:30












  • $begingroup$
    You might be able to use widely available printed tables of chi-squared distributions (with appriat modifications; halve DF to get gamma shape). Some printed tables have only the upper tail. Look around. If you're allergic to R, I suppose MatLab and such will do much the same computations. // But R is free, devotedly bug free for basic stuff like this, and trivial to install. See Wikipedia. // Code qchisq(.95, 4) and qgamma(.95, 2, 1/2) both return 9.487729
    $endgroup$
    – BruceET
    Dec 18 '18 at 1:08












  • $begingroup$
    Actually one can just take the $1-frac{alpha}{2}$-quantile by definition(?) or just calculate it like @BruceET did. :-)
    $endgroup$
    – user408858
    Dec 20 '18 at 0:46


















  • $begingroup$
    Of course, the integral in the end is the same as in the definition of the Gammafunction.
    $endgroup$
    – user408858
    Dec 18 '18 at 0:12










  • $begingroup$
    If I understand your notation and if you need the answers instead of hints how to do the calculus, then here is R code that will give you the answer for $n = 1, 2, dots, 50$ and $alpha = 0.05:$ n = 1:50; L = qgamma(.025, n, 2); U = qgamma(.975, n ,2); cbind(n, L, U). For $n = 1,$ you have the exponential distribution, for which the calculus is trivial. For $n > 200,$ you should get good answers with normal approximation.
    $endgroup$
    – BruceET
    Dec 18 '18 at 0:28












  • $begingroup$
    Oh! Do I get you right, that there are these tables in backmatter of statistic books for exactly these kind of questions? :-)
    $endgroup$
    – user408858
    Dec 18 '18 at 0:30












  • $begingroup$
    You might be able to use widely available printed tables of chi-squared distributions (with appriat modifications; halve DF to get gamma shape). Some printed tables have only the upper tail. Look around. If you're allergic to R, I suppose MatLab and such will do much the same computations. // But R is free, devotedly bug free for basic stuff like this, and trivial to install. See Wikipedia. // Code qchisq(.95, 4) and qgamma(.95, 2, 1/2) both return 9.487729
    $endgroup$
    – BruceET
    Dec 18 '18 at 1:08












  • $begingroup$
    Actually one can just take the $1-frac{alpha}{2}$-quantile by definition(?) or just calculate it like @BruceET did. :-)
    $endgroup$
    – user408858
    Dec 20 '18 at 0:46
















$begingroup$
Of course, the integral in the end is the same as in the definition of the Gammafunction.
$endgroup$
– user408858
Dec 18 '18 at 0:12




$begingroup$
Of course, the integral in the end is the same as in the definition of the Gammafunction.
$endgroup$
– user408858
Dec 18 '18 at 0:12












$begingroup$
If I understand your notation and if you need the answers instead of hints how to do the calculus, then here is R code that will give you the answer for $n = 1, 2, dots, 50$ and $alpha = 0.05:$ n = 1:50; L = qgamma(.025, n, 2); U = qgamma(.975, n ,2); cbind(n, L, U). For $n = 1,$ you have the exponential distribution, for which the calculus is trivial. For $n > 200,$ you should get good answers with normal approximation.
$endgroup$
– BruceET
Dec 18 '18 at 0:28






$begingroup$
If I understand your notation and if you need the answers instead of hints how to do the calculus, then here is R code that will give you the answer for $n = 1, 2, dots, 50$ and $alpha = 0.05:$ n = 1:50; L = qgamma(.025, n, 2); U = qgamma(.975, n ,2); cbind(n, L, U). For $n = 1,$ you have the exponential distribution, for which the calculus is trivial. For $n > 200,$ you should get good answers with normal approximation.
$endgroup$
– BruceET
Dec 18 '18 at 0:28














$begingroup$
Oh! Do I get you right, that there are these tables in backmatter of statistic books for exactly these kind of questions? :-)
$endgroup$
– user408858
Dec 18 '18 at 0:30






$begingroup$
Oh! Do I get you right, that there are these tables in backmatter of statistic books for exactly these kind of questions? :-)
$endgroup$
– user408858
Dec 18 '18 at 0:30














$begingroup$
You might be able to use widely available printed tables of chi-squared distributions (with appriat modifications; halve DF to get gamma shape). Some printed tables have only the upper tail. Look around. If you're allergic to R, I suppose MatLab and such will do much the same computations. // But R is free, devotedly bug free for basic stuff like this, and trivial to install. See Wikipedia. // Code qchisq(.95, 4) and qgamma(.95, 2, 1/2) both return 9.487729
$endgroup$
– BruceET
Dec 18 '18 at 1:08






$begingroup$
You might be able to use widely available printed tables of chi-squared distributions (with appriat modifications; halve DF to get gamma shape). Some printed tables have only the upper tail. Look around. If you're allergic to R, I suppose MatLab and such will do much the same computations. // But R is free, devotedly bug free for basic stuff like this, and trivial to install. See Wikipedia. // Code qchisq(.95, 4) and qgamma(.95, 2, 1/2) both return 9.487729
$endgroup$
– BruceET
Dec 18 '18 at 1:08














$begingroup$
Actually one can just take the $1-frac{alpha}{2}$-quantile by definition(?) or just calculate it like @BruceET did. :-)
$endgroup$
– user408858
Dec 20 '18 at 0:46




$begingroup$
Actually one can just take the $1-frac{alpha}{2}$-quantile by definition(?) or just calculate it like @BruceET did. :-)
$endgroup$
– user408858
Dec 20 '18 at 0:46










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