I want to work out the quantiles of a linear combiation of chi square random variables.

Suppose $\lambda_i \in \mathbb{R}$ for all $i \in \{1,2,\cdots,n\}$ and $Z = \sum_{i = 1}^n \lambda_iX_i$ and $X_1,X_2,\cdots,X_n \stackrel{\text{iid}}{\sim} \chi^2_1$.

How can I work out the quantiles?

  • $\begingroup$ The same question about the distributions is asked and answered at stats.stackexchange.com/questions/2035/…. Since "working out the quantiles" is tantamount to finding the distribution and a chi-squared distribution is a Gamma distribution, I take the other to be a duplicate of this one. $\endgroup$
    – whuber
    Aug 20, 2017 at 19:35
  • $\begingroup$ The Q is formulated as if the OP is also interested in negative $\lambda_i$, which is not covered by the linked post. $\endgroup$ Aug 22, 2017 at 23:42
  • $\begingroup$ @whuber anyhow, abettere duplicate is stats.stackexchange.com/questions/72479/… $\endgroup$ Aug 23, 2017 at 21:14
  • 1
    $\begingroup$ @Kjetil Thank you for pointing that out. I chose the duplicate I did because it deals with linear combinations rather than just sums. Nevertheless, your information is valuable and so I will include that thread within the list of duplicates. The system enables us to list up to five duplicates for any closed question, so feel free to identify more! $\endgroup$
    – whuber
    Aug 23, 2017 at 22:10


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