Probability density function (PDF) of a scaled non-central chi squared distribution












0












$begingroup$


I know that:




  1. If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.


  2. If $X$ is central chi-squared distributed with $nu$ degree of freedom, then $cX$ is gamma distributed with shape $frac{nu}{2}$ and scale parameter $2c$.



where $c$ is a positive constant.
My question is:



If $X$ is non-central chi-squared distributed with $nu$ degree of freedom, then how is $cX$ distributed? What is the pdf of $cX$ ?










share|cite|improve this question









$endgroup$












  • $begingroup$
    In general, the PDF of $cX$ is $frac{1}{c} f_X(t/c)$ where $f_X$ is the PDF of $X$. I don't know of a nice name for the distribution of your particular $cX$ though.
    $endgroup$
    – angryavian
    Jan 1 at 5:48










  • $begingroup$
    Thank you @angryavian. The information you gave is very helpful. By the way, is there a reference where the PDF of $cX$ is $frac{1}{c}f_X(frac{t}{c}) $?
    $endgroup$
    – M.A.N
    Jan 1 at 7:07










  • $begingroup$
    This is a special case of change of variables.
    $endgroup$
    – angryavian
    Jan 1 at 17:22
















0












$begingroup$


I know that:




  1. If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.


  2. If $X$ is central chi-squared distributed with $nu$ degree of freedom, then $cX$ is gamma distributed with shape $frac{nu}{2}$ and scale parameter $2c$.



where $c$ is a positive constant.
My question is:



If $X$ is non-central chi-squared distributed with $nu$ degree of freedom, then how is $cX$ distributed? What is the pdf of $cX$ ?










share|cite|improve this question









$endgroup$












  • $begingroup$
    In general, the PDF of $cX$ is $frac{1}{c} f_X(t/c)$ where $f_X$ is the PDF of $X$. I don't know of a nice name for the distribution of your particular $cX$ though.
    $endgroup$
    – angryavian
    Jan 1 at 5:48










  • $begingroup$
    Thank you @angryavian. The information you gave is very helpful. By the way, is there a reference where the PDF of $cX$ is $frac{1}{c}f_X(frac{t}{c}) $?
    $endgroup$
    – M.A.N
    Jan 1 at 7:07










  • $begingroup$
    This is a special case of change of variables.
    $endgroup$
    – angryavian
    Jan 1 at 17:22














0












0








0





$begingroup$


I know that:




  1. If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.


  2. If $X$ is central chi-squared distributed with $nu$ degree of freedom, then $cX$ is gamma distributed with shape $frac{nu}{2}$ and scale parameter $2c$.



where $c$ is a positive constant.
My question is:



If $X$ is non-central chi-squared distributed with $nu$ degree of freedom, then how is $cX$ distributed? What is the pdf of $cX$ ?










share|cite|improve this question









$endgroup$




I know that:




  1. If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.


  2. If $X$ is central chi-squared distributed with $nu$ degree of freedom, then $cX$ is gamma distributed with shape $frac{nu}{2}$ and scale parameter $2c$.



where $c$ is a positive constant.
My question is:



If $X$ is non-central chi-squared distributed with $nu$ degree of freedom, then how is $cX$ distributed? What is the pdf of $cX$ ?







statistics probability-distributions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 1 at 4:18









M.A.NM.A.N

1028




1028












  • $begingroup$
    In general, the PDF of $cX$ is $frac{1}{c} f_X(t/c)$ where $f_X$ is the PDF of $X$. I don't know of a nice name for the distribution of your particular $cX$ though.
    $endgroup$
    – angryavian
    Jan 1 at 5:48










  • $begingroup$
    Thank you @angryavian. The information you gave is very helpful. By the way, is there a reference where the PDF of $cX$ is $frac{1}{c}f_X(frac{t}{c}) $?
    $endgroup$
    – M.A.N
    Jan 1 at 7:07










  • $begingroup$
    This is a special case of change of variables.
    $endgroup$
    – angryavian
    Jan 1 at 17:22


















  • $begingroup$
    In general, the PDF of $cX$ is $frac{1}{c} f_X(t/c)$ where $f_X$ is the PDF of $X$. I don't know of a nice name for the distribution of your particular $cX$ though.
    $endgroup$
    – angryavian
    Jan 1 at 5:48










  • $begingroup$
    Thank you @angryavian. The information you gave is very helpful. By the way, is there a reference where the PDF of $cX$ is $frac{1}{c}f_X(frac{t}{c}) $?
    $endgroup$
    – M.A.N
    Jan 1 at 7:07










  • $begingroup$
    This is a special case of change of variables.
    $endgroup$
    – angryavian
    Jan 1 at 17:22
















$begingroup$
In general, the PDF of $cX$ is $frac{1}{c} f_X(t/c)$ where $f_X$ is the PDF of $X$. I don't know of a nice name for the distribution of your particular $cX$ though.
$endgroup$
– angryavian
Jan 1 at 5:48




$begingroup$
In general, the PDF of $cX$ is $frac{1}{c} f_X(t/c)$ where $f_X$ is the PDF of $X$. I don't know of a nice name for the distribution of your particular $cX$ though.
$endgroup$
– angryavian
Jan 1 at 5:48












$begingroup$
Thank you @angryavian. The information you gave is very helpful. By the way, is there a reference where the PDF of $cX$ is $frac{1}{c}f_X(frac{t}{c}) $?
$endgroup$
– M.A.N
Jan 1 at 7:07




$begingroup$
Thank you @angryavian. The information you gave is very helpful. By the way, is there a reference where the PDF of $cX$ is $frac{1}{c}f_X(frac{t}{c}) $?
$endgroup$
– M.A.N
Jan 1 at 7:07












$begingroup$
This is a special case of change of variables.
$endgroup$
– angryavian
Jan 1 at 17:22




$begingroup$
This is a special case of change of variables.
$endgroup$
– angryavian
Jan 1 at 17:22










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