For each distribution listed below, plot the corresponding characteristic function of the density as a function of \(t\) if the characteristic function is real-valued, or as a function of \(t\) on the complex plane if the function is complex-valued. Describe each characteristic function. Are there any properties of the associated random variables that have an apparent effect on the properties of the characteristic function? See Section B. 3 for details on plotting complex functions in the complex plane.

a. BERNOULLi \(\left(\frac{1}{2}ight)\)

b. \(\operatorname{BinOmial}\left(5, \frac{1}{2}ight)\)

c. \(\operatorname{GeOmetric}\left(\frac{1}{2}ight)\)

d. \(\operatorname{Poisson}(2)\)

e. \(\operatorname{Uniform}(0,1)\)

f. Exponential (2)

g. \(\operatorname{Cauchy}(0,1)\)