Problem 1
Let \(\mu\) be an arbitrary finite o-additive measure on the real line \((-\infty, c o)\). Represent \(\mu\) as the Stieltjes measure corresponding to some generating function \(F\). Hint. Let \(F(x)=\mu(-\mathrm{a}, x)\). Comment. Thus the term "Stieltjes measure" does not refer to a special kind of measure, but rather to a special way of constructing a measure (by using a generating function).
Problem 4
Find the mean and variance of the random variable \(\xi\) with probability
density
$$
p(x)=\frac{1}{2} e^{-|x|} \quad(-\infty
Problem 5
Let \(\xi\) be the random variable with probability density
$$
p(x)=\frac{1}{\pi\left(1+x^{2}\right)} \quad(-\infty
Problem 6
Discuss random varables which are neither discrete nor continuous.
Problem 9
Write formulas for the Riemann-Stieltjes integral (40) in the case where \(\mathbf{f}\) is continuous and a) \(\Phi\) is a jump function; b) \(\Phi\) is an absolutely continuous function with a Riemann-integrable derivative.
Problem 10
Evaluate the following Riemann-Stieltjes integrals:
a) \(\int_{-1}^{3} x d F(x)\), where \(F(x)=\left\\{\begin{aligned} 0 & \text {
if } x=-1 \\ 1 & \text { if }-1
