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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 …

Streaks of heads and tails