minimum of chi-squared distribution

[Preeya27]

Hi

I have a query on finding the minimum of a chi-squared distribution (M).

I have attached my working so far. But I have difficulty evaluating the density function of M. How could this be reduced to a simplified form

Any help would be much appreciated.

Thank you

 

[Dason]

Did somebody tell you that this could be solved for analytically?

 

[Preeya27]

No. Can it be solved for analytically?

 

[Dason]

I don't think so. But it can be done numerically.

 

[Preeya27]

Could you show me how to go about this?

 

[Dason]

What part is giving you trouble

 

[Dason]

What software are you using

 

[Preeya27]

I am not using any software; I am trying to do this manually

 

[Dason]

Well one suggestion would be that you don't need to try to evaluate the pdf at all. Just consider the CDF. What would the cdf look like in terms of the cdf of the original distribution?

 

[Dason]

You don't need the pdf. Start from the definition of Cdf. You can figure out the cdf of the min with respect to the cdf of the original random variable with no need to calculate the pdf.