In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $( a , b ) \in \mathbf { R } ^ { 2 }$ and all real $\theta$,
$$\Phi _ { a X + b } ( \theta ) = \frac { p e ^ { i ( a + b ) \theta } } { 1 - q e ^ { i a \theta } }$$