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$X\sim \chi^2(n)$ with probability density function $f(x)$.
$Y\sim \chi^2(n+2)$ with probability density function $g(x)$.

How to determine a search procedure (or algorithm) to find the upper bound $c_1$ and lower bound $c_2$ so that $$ \int_{c_1}^{c_2} f(x)dx=\int_{c_1}^{c_2} g(x)dx\text{?} $$

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1 Answer 1

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Let $c$ be a root to $f(x)-g(x)$, such that $f(c)=g(c)$. Note that $c_1$ and $c_2$ must be on either side of $c$. Start with a guess $c_1=c-\epsilon$, $c_2=c+\epsilon$, evaluate the integral, and adjust $c_1$ and $c_2$ until the desired precision is reached.

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