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Posts from the ‘Probability Theory’ Category

Oct 19

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.

Sep 25

8 Comments

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.

Aug 28

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}.

Aug 25

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 …

Streaks of heads and tails