Probability Measures

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $A \in \mathcal{F}$. Prove that $A$ and $\Omega \backslash A$ are independent if and only if $\mathbb {P}\left(A\right) = 0$ or $\mathbb {P}\left(A\right) = 1$.

Suppose $\mathbb {P}$ is Lebesgue measure on [0,1]. Give an example of two disjoint Borel subsets $A$ and $B$ of $[0,1]$ such that $\mathbb {P}\left(A\right) = \mathbb {P}\left(B\right) = \frac{1}{2},\left[0, \frac{1}{2}\right]$ and $A$ are independent, and $\left[0, \frac{1}{2}\right]$ and $B$ are independent.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $A, B \in \mathcal{F}$. Prove that the following are equivalent:

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $\left\{ A_{k}\right\}_{k \in \Gamma }$ is a family of events. Prove the family $\left\{ A_{k}\right\}_{k \in \Gamma }$ is independent if and only if the family $\left\{ \Omega \backslash A_{k}\right\}_{k \in \Gamma }$ is independent.

Give an example of a probability space $(\Omega , \mathcal{F}, \mathbb {P})$ and events $A, B_{1}, B_{2}$ such that $A$ and $B_{1}$ are independent, $A$ and $B_{2}$ are independent, but $A$ and $B_{1} \cup B_{2}$ are not independent.

Give an example of a probability space $(\Omega , \mathcal{F}, \mathbb {P})$ and events $A_{1}, A_{2}, A_{3}$ such that $A_{1}$ and $A_{2}$ are independent, $A_{1}$ and $A_{3}$ are independent, and $A_{2}$ and $A_{3}$ are independent, but the family $A_{1}, A_{2}, A_{3}$ is not independent.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space, $A \in \mathcal{F}$, and $B_{1} \subseteq B_{2} \subseteq \cdots$ is an increasing sequence of events such that $A$ and $B_{n}$ are independent events for each $n \in \mathbb {N}$. Show that $A$ and $\bigcup_{n = 1}^{\infty } B_{n}$ are independent.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $\left\{ A_{t}\right\}_{t \in \mathbb {R}}$ is an independent family of events such that $\mathbb {P}\left(A_{t}\right) < 1$ for each $t \in \mathbb {R}$. Prove that there exists a sequence $t_{1}, t_{2}, \ldots$ in $\mathbb {R}$ such that $\mathbb {P}\left(\bigcap_{n = 1}^{\infty } A_{t_{n}}\right) = 0$.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $B_{1}, \ldots , B_{n} \in \mathcal{F}$ are such that $\mathbb {P}\left(B_{1} \cap \cdots \cap B_{n}\right) > 0$. Prove that $\mathbb {P}\left(A \cap B_{1} \cap \cdots \cap B_{n}\right) = \mathbb {P}\left(B_{1}\right) \cdot \mathbb {P}_{B_{1}}\left(B_{2}\right) \cdots \mathbb {P}_{B_{1} \cap \cdots \cap B_{n-1}}\left(B_{n}\right) \cdot \mathbb {P}_{B_{1} \cap \cdots \cap B_{n}}\left(A\right)$ for every event $A \in \mathcal{F}$.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $A \in \mathcal{F}$ is an event such that $0 < \mathbb {P}\left(A\right) < 1$. Prove that

for every event $B \in \mathcal{F}$.

Give an example of a probability space $(\Omega , \mathcal{F}, \mathbb {P})$ and $X, Y \in \mathcal{L}^{2}(P)$ such that $\operatorname {Var}\left[X+Y\right] = \operatorname {Var}\left[X\right]+\operatorname {Var}\left[Y\right]$ but $X$ and $Y$ are not independent random variables.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ and $\left(\Omega^{\prime }, \mathcal{F}^{\prime }, \mathbb {P}^{\prime }\right)$ are probability spaces, $X$ is a random variable on $\Omega , Y$ is a random variable on $\Omega^{\prime }$, and $\widetilde{X} = \widetilde{Y}$. Prove that $\mathbb {P}_{X} = \mathbb {P}_{Y}^{\prime }$.

Suppose $H: \mathbb {R} \rightarrow (0,1)$ is a continuous one-to-one function satisfying conditions (a) through (d) of 12.29. Show that the function $X:(0,1) \rightarrow \mathbb {R}$ produced in the proof of 12.29 is the inverse function of $H$.

Suppose $(\Omega , \mathcal{F}, \mathbb {P})$ is a probability space and $X$ is a random variable. Prove that the following are equivalent:

Suppose $\alpha > 0$ and $h(x) = \begin{cases} 0 & \text{ if } x < 0, \\ \alpha^{2} x e^{-\alpha x} & \text{ if } x \geq 0.\end{cases}$ Let $\mathbb {P} = h d \lambda$ and let $X$ be the random variable defined by $X(x) = x$ for $x \in \mathbb {R}$.

Suppose $\mathcal{B}$ is the $\sigma$-algebra of Borel subsets of $[0,1)$ and $\mathbb {P}$ is Lebesgue measure on $([0,1), \mathcal{B})$. Let $\left\{ e_{k}\right\}_{k \in \mathbb {N}}$be the family of functions defined by the fourth bullet point of Example 8.51 (notice that $k = 0$ is excluded). Show that the family $\left\{ e_{k}\right\}_{k \in \mathbb {N}}$is an i.i.d.

Suppose $\mathcal{B}$ is the $\sigma$-algebra of Borel subsets of $(-\pi , \pi ]$ and $\mathbb {P}$ is Lebesgue measure on $((-\pi , \pi ], \mathcal{B})$ divided by $2 \pi$. Let $\left\{ e_{k}\right\}_{k \in \mathbf{Z} \backslash \left\{ 0\right\} }$ be the family of trigonometric functions defined by the third bullet point of Example 8.51 (notice that $k = 0$ is excluded).