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Let $$s_i \sim \chi(k_i, \lambda_i), i\in \{ 1, 2\}$$ be two non-central chi-squared random variables with $k_i$ degrees of freedom and $\lambda_i$ parameter of non-centrality I am wondering if $$p_{max} = max(s_1,s_2), p_{min} = min(s_1,s_2)$$ are also non-central chi squared random variables? What I quickly did was to implement it in MATLAB and I realized that both $$p_{max} , p_{min}$$ follow non central chi-squared distributions but I don't have a concrete proof of it. Also, if my assumption is true, what would be the relationships between $ k_i,\lambda_i$ and parameters for both of these $p_{max}, p_{min}$. If this is a known problem, please refer me to a text or similar because I have not found this problem solved anywhere.

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up vote 2 down vote accepted

Let $X$ and $Y$ be two independent random variables. Then: $$ F_{\max(X,Y)}(z) = \mathbb{P}\left(\max(X,Y) \leqslant z\right) = \mathbb{P}(X \leqslant z, Y \leqslant z) = \mathbb{P}(X \leqslant z) \mathbb{P}(Y \leqslant z) = F_X(z) F_Y(z) $$ Similarly: $$ 1- F_{\min(X,Y)}(z) = \mathbb{P}\left(\min(X,Y)> z\right) = \mathbb{P}(X > z, Y > z) = \mathbb{P}(X > z) \mathbb{P}(Y > z) = (1-F_X(z))(1-F_Y(z)) $$ Since the product of cumulative distribution functions of two non-central $\chi^2$ random variables does not have the functional form of the cumulative distribution of non-central $\chi^2$ random variable, your conclusion is premature. Neither $\min(X,Y)$ nor $\max(X,Y)$ follow a non-central $\chi^2$ distribution.

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