\sum\limits_{x = 1}^\infty {xe^{ - \lambda } \lambda \frac{{\lambda ^{x - 1} }}
{{\left( {x - 1} \right)!}}} = e^{ - \lambda } \lambda \sum\limits_{x = 1}^\infty {x\frac{{\lambda ^{x - 1} }}
{{\left( {x - 1} \right)!}}} = e^{ - \lambda } \lambda \sum\limits_{x = 1}^\infty {\left( {x - 1 + 1} \right)\frac{{\lambda ^{x - 1} }}
{{\left( {x - 1} \right)!}}} =