Math Problems Vault

Problem 8.1

(a)(b)(c)

Let $\xi$ and $\eta$ be random variables. Prove that the following sets are events:

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(a)

$A = \left\{ \omega \in \Omega : \xi (\omega ) < \eta (\omega )\right\}$ ;

(b)

$B = \left\{ \omega \in \Omega : \xi (\omega ) \leqslant \eta (\omega )\right\}$ ;

(c)

$C = \left\{ \omega \in \Omega : \xi (\omega ) = \eta (\omega )\right\}$ .

Problem 8.2

Let the function $\xi : \Omega \rightarrow \mathbb {R}$ be such that for every $x \in \mathbb {R}$ the set $A_x = \left\{ \omega \in \Omega : \xi (\omega ) = x\right\}$ is an event. Is it true that $\xi$ must then be a random variable?

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Problem 8.3

Construct a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and a function $\xi : \Omega \rightarrow \mathbb {R}$ on it such that $\xi$ is not a random variable.

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Problem 8.4

(a)(b)(c)(d)

Must the function $\xi$ be a random variable if the following function is a random variable:

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(a)

$\eta = \xi^2$ ;

(b)

$\eta = e^\xi$ ;

(c)

$\eta = \left|\xi \right|$ ;

(d)

$\eta = \xi + 1$ .

Problem 8.5

(a)(b)(c)(d)

Let the probability space $(\Omega , \mathscr {F}, \mathbb {P})$ be the segment $[0, 1]$ with the $\sigma$ -algebra of Borel sets and Lebesgue measure. Describe the $\sigma$ -algebra generated by the random variable:

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(a)

$\xi (\omega ) = 1/2$ ;

(b)

$\xi (\omega ) = \begin{cases} \omega , & \omega \in [0, 1/2), \\ 1, & \omega \in [1/2, 1]; \end{cases}$

(c)

$\xi (\omega ) = \omega /2$ ;

(d)

$\xi (\omega ) = \begin{cases} 1/4, & \omega \in [0, 1/4), \\ 1/2, & \omega \in [1/4, 3/4), \\ 1, & \omega \in [3/4, 1]. \end{cases}$

Problem 8.6

(a)(b)

Let the probability space $(\Omega , \mathscr {F}, \mathbb {P})$ be the line $\mathbb {R}$ with the $\sigma$ -algebra of Borel sets and some probability measure. Describe the $\sigma$ -algebra generated by the random variable:

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(a)

$\xi (\omega ) = \omega^2$ ;

(b)

$\xi (\omega ) = \cos \omega$ .