Assume $X_1$ is an Exponential random variable with unit mean ( i.e. $f_{X_1}(x) = e^{-x}$ ) and $X_2$ is an Erlang distribution with shape $N$ and unit rate ( i.e. $f_{X_2}(x) = \frac{x^{N-1}e^{-x}}{(N-1)!}$ ).
I need to compute the expected value of $X=\frac{X_1+X_1X_2}{c+\alpha X_1+\beta X_1X_2}$ where $c$, $\alpha$ and $\beta$ are larger than zero.
I tried to find the expectation by calculating $f_X(x)$ and then $\int_0^\infty xf_X(x)dx$, but I could not calculate integral.
I also tried $E(X)=\int_0^\infty P[X>x] dx$, but again failed calculating $E(X)$.
I am not even sure if $E(X)$ is available in closed form at all or not. I would appreciate if one could help me.
Best Answer
By definition, $$ \mathrm E(X)=\iint_{x\geqslant0,z\geqslant0}\frac{x(1+z)}{c+x(\alpha+\beta z)}\mathrm e^{-x}\frac{z^{N-1}}{(N-1)!}\mathrm e^{-z}\mathrm dx\mathrm dz, $$ which is equivalent to $$ \mathrm E(X)=\iint_{x\geqslant0,x_i\geqslant0}\frac{x(1+x_1+\cdots+x_N)}{c+x(\alpha+\beta(x_1+\cdots+x_N))}\mathrm e^{-x-x_1-\cdots-x_N}\mathrm dx\mathrm dx_1\cdots\mathrm dx_N. $$