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I was working with gamma random variables. Let

$Y_1$ the maximum of $N_1$ iid gamma random variables with parameter $m_1$ and $\beta_1$.

Similar let $Y_2$ the maximum of $N_2$ iid gamma random variables with parameter $m_2$ and $\beta_2$.

The PDF and CDF of $Y_i$ are

\begin{align} f_{Y_i}(y)=&\frac{N_i(m_i\beta_i)^{m_i}} {\Gamma(m_{i})^{N_i}}\Big[\gamma(m_{i},m_{i}y\beta_{i})\Big]^{N_i-1}y^{m_{i}-1}e^{-m_{i}y\beta_{i}}\\ F_{Y_i}(y)=& \left[\frac{\gamma(m_i,m_i\beta_i y)}{\Gamma(m_i)}\right]^{N_i} \end{align}

Also I found the following representation

\begin{align}\label{} f_{Y_i}(y)=&N_i\sum_{n_i=0}^{N_i-1}\sum_{k_i=0}^{n_i(m_i-1)}\frac{\binom{N_i-1}{n_i}}{\Gamma(m_i)}(-1)^{n_i} (\beta_i m_i)^{k_i+m_i}y^{m_i+k_i-1}e^{-m_i(n_i+1)\beta_i y}\phi_{k_i,n_i,m_i} \\ F_{Y_i}(y)=&\sum_{n_i=0}^{N_i}\sum_{k_i=0}^{n_i(m_i-1)}\binom{N_i}{n_i}(-1)^{n_i}e^{-m_in_i\beta_i y}\phi_{k_i,n_i,m_i}\left(m_i\beta_i y\right)^{k_i}. \end{align}.

For small $\beta_i$ I have found the following PDF and CDF of $Y_i$

\begin{align}\label{} F_{Y_{i}}(y)&\approx\left[\frac{(m_i\beta_{i}y)^{m_i}}{m_i!}\right]^{N_{i}}\\ \label{pcdf_ap} f_{Y_i}(y)&\approx m_iN_{i}\left[\frac{(m_i\beta_{i})^{m_i}}{m_i!}\right]^{N_{i}}y^{m_iN_{i}-1}. \end{align}

Only two question I have, I didn't understand how we get the PDF and CDF approximation.

They said using \begin{align}\label{Y_c_pdf} F_{Y_i}(y)=& \left[\frac{\gamma(m_i,m_i\beta_i y)}{\Gamma(m_i)}\right]^{N_i}=\left[1-e^{-m_i\beta_i y}\sum_{k=0}^{m_i-1}\left(m_i\beta_i y\right)^k \frac{1}{k!}\right]^{N_i} \end{align} then using $e^{-\alpha}=1-\alpha$ and employing the lowest order terms corresponding in Low SNR regime $\beta_i$. Now I am using this approximation and it work.

But When I am try to simplify the following probability using PDF and CDF approximation

\begin{align} P\{Y_1\leq \alpha\leq Y_2\}&=\int_{y_1=0}^{\alpha}f_{Y_1}(y_1)dy_1 \int_{y_2=\alpha}^{\infty}f_{Y_2}(y_2)dy_2\\ &=F_{Y_1}(\alpha)\left[1-\int_{y_2=\alpha}^{\infty}f_{Y_2}(y_2)dy_2\right]\\ &=F_{Y_1}(\alpha)\left[1-F_{Y_2}(\alpha)\right] \end{align} I would like to know if there is any possible approximation for

$$1-F_{Y_2}(\alpha)=1-\left[\frac{\gamma(m_i,m_i\beta_i y)}{\Gamma(m_i)}\right]^{N_i} $$

to get equation similar to the CDF approximation?

Thanks.

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