complex combination of a bernoulli random variable

Question

I would like to know what is the characteristic function of the follow complex combination of a Bernoulli random variable. Consider $$d= \Bigg\{\begin{array}{lcl} 0 & \text{with probability} & 1-p\\ 1 & \text{with probability} & p\\ \end{array} $$ a Bernoulli random variable and let $ z \in \mathbb{C}$, what is the characteristic function $\phi_{zd}(t)$? Thanks in advance!.

Answer 1

For a complex valued random variable $X$ the characteristic function $\phi_X$ is defined as the mapping $$\mathbb C \ni t \mapsto \phi_X(t) := E[\exp(i\;\text{Re} (\bar{t}X))] \in \mathbb C.$$ For a reference, see Wikipedia for instance.

Now, in our case $$\begin{align}\phi_{zd}(t)=E[\exp(i\;\text{Re}(\bar{t}zd))]&=p\;\exp(i\;\text{Re}(\bar{t}z)) + (1-p)\;\exp(i\;\text{Re}(\bar{t} \cdot 0))\\&= p\;\exp(i\;\text{Re}(\bar{t}z)) + (1-p).\end{align}$$

Edit: You're absolutely right! I missed the second summand.

If you identify $\mathbb C$ with $\mathbb R ^2$ via $$\mathbb C \ni z \leftrightarrow (z_1,z_2) := (\text{Re}(z),\text{Im}(z)) \in \mathbb R^2$$ then $$\text{Re} (\bar{t}z)=t_1z_1+t_2z_2= \;<t,z>$$ with $<\cdot,\cdot>$ being the standard scalar product in $\mathbb R^2$.