Sum of Chi-Square Random Variables

Abstract

Define the RV Z₂=−Y₂. Then the PDF of Z₂ is given by $P_{Z_2 } \left( z \right) = P_{Y_2 } \left( { - z} \right),z \leqslant 0$. From the form of p_Y(y) for central chi-square RVs, we observe that for n odd, the PDF of Z₂ is given by the PDF of Y₂, with y replaced by z and −σ ²₂ substituted for σ ²₂ . For n even, the PDF of Z₂ is given by the negative of the PDF of Y₂ with y replaced by z and −σ ²₂ substituted for σ ²₂ . From the form of p_Y(y) for noncentral chi-square RVs, we observe that in addition to the above substitutions, −σ ²₂ must be substituted for a ²₂ . For example, for Y₂ a central chi-square RV with 2m₂ degrees of freedom, the PDF of Z₂ is expressible as

$$ \begin{gathered} p_{Z_2 } \left( z \right) = p_{Y_2 } \left( { - z} \right) = \frac{1} {{2^{m_2 } \left( {\sigma _2^2 } \right)^{m_2 } \Gamma \left( {m_2 } \right)}}\left( { - z} \right)^{m_2 - 1} \exp \left( {\frac{z} {{2\sigma _2^2 }}} \right) \hfill \\ = \frac{1} {{2^{m_2 } \left( { - \sigma _2^2 } \right)^{m_2 } \Gamma \left( {m_2 } \right)}}z^{m_2 - 1} \exp \left( { - \frac{z} {{2\left( { - \sigma _2^2 } \right)}}} \right) \hfill \\ = - p_{Y_2 } \left( z \right)\left| {_{\sigma _2^2 \to - \sigma _2^2 } } \right.,z \leqslant 0 \hfill \\ \end{gathered} $$

(5.1)

that is, we use the expression for the PDF of Y₂ (which applies for y≥0) but substitute z for y, −σ ²₂ for σ ²₂ , and then take its negative and apply it for z ≤ 0. Similarly, for Y₂ a noncentral chi-square RV with 2m₂ degrees of freedom, the PDF of Z₂ is expressible as

$$ \begin{gathered} p_{Z_2 } \left( z \right) = \frac{1} {{2\sigma _2^2 }}\left( {\frac{{ - z}} {{a_2^2 }}} \right)^{{{\left( {m_2 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {m_2 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \exp \left( { - \frac{{ - z + a_2^2 }} {{2\sigma _2^2 }}} \right){\rm I}_{m_2 - 1} \left( {\sqrt {\frac{{a_2^2 \left( { - z} \right)}} {{\sigma _2^4 }}} } \right) \hfill \\ = - \frac{1} {{2\left( { - \sigma _2^2 } \right)}}\left( {\frac{z} {{ - a_2^2 }}} \right)^{{{\left( {m_2 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {m_2 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \exp \left( { - \frac{{z - a_2^2 }} {{2\left( { - \sigma _2^2 } \right)}}} \right){\rm I}_{m_2 - 1} \left( {\sqrt {\frac{{ - a_2^2 z}} {{\left( { - \sigma _2^2 } \right)^2 }}} } \right) \hfill \\ = - p_{Y_2 } \left( z \right)\left| {_{_{a_2^2 \to - a_2^2 }^{\sigma _2^2 \to - \sigma _2^2 } } } \right.,z \leqslant 0 \hfill \\ \end{gathered} $$

(5.2)