Probability density function (PDF) of a scaled non-central chi squared distribution
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I know that:
If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.
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
asked Jan 1 at 4:18
M.A.NM.A.N
1078
$endgroup$
add a comment |
$begingroup$
I know that:
If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.
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
asked Jan 1 at 4:18
M.A.NM.A.N
1078
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
I know that:
If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.
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
asked Jan 1 at 4:18
M.A.NM.A.N
1078
$endgroup$
I know that:
If $X$ is exponentially distributed with a rate parameter $lambda$, then $cX$ is exponentially distributed with a rate parameter $frac{lambda}{c}$.
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
statistics probability-distributions
asked Jan 1 at 4:18
M.A.NM.A.N
1078
asked Jan 1 at 4:18
M.A.NM.A.N
1078
asked Jan 1 at 4:18
M.A.NM.A.N
1078
asked Jan 1 at 4:18
M.A.NM.A.N
1078
asked Jan 1 at 4:18
M.A.NM.A.N
1078
1078
$begingroup$
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
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