# Posts from the ‘Probability Theory’ Category

## Homework 4, due Friday Oct 24

Jacod and Protter, problems 11.5, 12.4 and 12.11

11.15 Let $F$ be a continuous distribution function and the $U$ be uniform on $(0,1)$. Define $G(u) = \inf \{x \, : \, F(x) \geq u\}$. Show that $G(u)$ has distribution function $F$.

12.4 Let $\rho_{X,Y}$ denote the correlation coefficient for $(X,Y)$. Let $a > 0$, $c > 0$ and $b \in \mathbb{R}$. Show that $\displaystyle \rho_{aX + b,cY+b} = \rho_{X,Y}$.

(This is useful because it shows that $\rho$ is independent of the scale of measurement for $X$ and $Y$.)

12.11 Let $(X,Y)$ be independent normals, both with means $\mu = 0$ and variances $\sigma^2$. Let $\displaystyle Z = \sqrt{X^2 + Y^2}$ and $\displaystyle W = \arctan\Big(\frac{X}{Y}\Big)$

where $-\frac{\pi}{2} \leq W \leq \frac{\pi}{2}$. Show that $Z$ has a Rayleigh distribution, that $W$ is uniform on $(-\frac{\pi}{2},\frac{\pi}{2})$, and $Z$ and $W$ are independent.

## Probability Theory, Homework 3, due Friday Oct 3.

From Jacod and Protter

9.5  Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Suppose that $X$ is a random variable with $X \geq 0$ almost surely and $\mathbb{E}(X) = 1$. Define $Q : \mathcal{A} \to \mathbb{R}$ by $Q(A) = \mathbb{E}(X 1_A)$. Show that $Q$ defines a probability measure on $(\Omega, \mathcal{A})$.

9.7 Suppose that $\mathbb{P}(X > 0) = 1$ and let $Q$ be defined as above. Let $\mathbb{E}^Q$ denote expectation with respect to $Q$.  Show that $\mathbb{E}^Q(Y) = \mathbb{E}^P(Y X)$.

Further exercises

Exercise 3. Suppose that $X \geq 0$ and $Y \geq 0$ are random variables and that $p \geq 0$.

1. Prove $\displaystyle \mathbb{E}[(X + Y)^p] \leq 2^p \big(\mathbb{E}(X^p) + \mathbb{E}(Y^p)\big)$.
2. If $p > 1$, the factor $2^p$ may be replaced with $2^{p-1}$.
3. If $0 \leq p \leq 1$, the factor $2^p$ can be replaced with 1.

Exercise 4. Suppose that $\mathbb{E}(X^2) = 1$ and $\mathbb{E}(|X|) \geq a > 0$. Prove for $0 \leq \lambda \leq 1$ that

CORRECTION: $\displaystyle \mathbb{P}\big(|X| \geq \lambda a\big) \geq (1 - \lambda)^2 a^2$.

Hint:  Think Cauchy-Schwarz.

## Probability Theory: Homework 1

Exercises, due Wed Sept 3.

From Jacod and Protter, problems 2.6, 2.7, 2.10 and 2.14.

2.6 Let $\mathcal{A}$ be a $\sigma$-algebra of subsets of $\Omega$ and let $B \in \mathcal{A}$. Show that $\mathcal{F} := \{A \cap B \, : \, A \in \mathcal{A}\}$ is a $\sigma$-algebra of subsets of $B$.  Is this still true if $B$ is a subset of $\Omega$ that is not a member of $\mathcal{A}$?

2.7  Let $f$ be a function mapping $\Omega$ to another space $E$ with a $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{A} := \{A \subset \Omega \, : \, \text{there exists } B \in \mathcal{E} \text{ with } A = f^{-1}(B)\}$. Show that $\mathcal{A}$ is a $\sigma$-algebra on $\Omega$.

2.10 Using the definition of a probability measure, show that if $A, \, B\, \in \mathcal{A}$, then $\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)$.

2.14 Suppose that $\mathbb{P}(A) = \frac{3}{4}$ and $\mathbb{P}(B) = \frac{1}{3}$. Show that $\frac{1}{12} \leq \mathbb{P}(A \cap B) \leq \frac{1}{3}$.

## Welcome to Probability and Measure Theory

In our first class we asked the question “What is randomness?” and started thinking about the Axiomatic Foundation of Probability. More details to come … 