Chapter 2.3, Problem 20E

Construct a sequence of squares in the first quadrant with one vertex at the origin and having sides of length $1-\frac{1}{2}x,x=1,2,\mathrm{....}$.

Select a point randomly from the unit square with one vertex at the origin and sides of length one. Let the random variable X equal x if the point is in the region between the squares with sides of lengths $1-\frac{1}{2^x}$ and $1-\frac{1}{2^{x-1}}$, x = 1,2,3,....

(a) Draw a figure illustrating this exercise.

(b) Show that $\begin{array}{l}f\left(x\right)=P\left(X=x\right)={\left(1-\frac{1}{2^x}\right)}^2-{\left(1-\frac{1}{2^{x-1}}\right)}^2\\ =\frac{2^{x+1}-3}{2^{2x}},x=1,2,3,\mathrm{...}\end{array}$

(c) Show that f(x) is a pmf.

(d) Find the moment-generating function of X.

(e) Find the mean of X.

(f) Find the variance of X.