19
Paper 2, Section I
3F Probability
Let U be a uniform random variable on (0, 1), and let > 0.
(a) Find the distribution of the random variable (log U )/.
(b) Dene a new random variable X as follows: suppose a fair coin is tossed, and if it
lands heads w

Poisson Processes
1
Introduction
Poisson processes are continuous time stochastic processes dened on the
time interval [0, ) with state space S = cfw_0, 1, 2, . It is a counting process whose state at time t represents the total number of certain
events t

Chapter 1
Introduction to Stochastic Processes
Exercise 1.1 The number of signals originated from a source in the interval (0, t], N (t), has
a Poisson (t) distribution. Each signal is registered by a receptor with probability p, independently of what hap

Chapter 1
Introduction to Stochastic Processes
Exercise 1.1 The number of signals originated from a source in the interval (0, t], N (t), has
a Poisson (t) distribution. Each signal is registered by a receptor with probability p, independently of what hap

Lecture 6.5
In this lecture Poisson processes are derived from rst principles and the notion of Markov chains
is extended to include continuous time Markov chains. This is necessary for our original aim of
modelling the M/M/1 queue with Markov chains (rem

Chapter 4
Markov Chains
Markov chains are an important tool for modeling stochastic processes that occur frequently in computer science. For example, after introducing the basic theory, we will
demonstrate how they are used for deriving randomized algorit

1
Notes on Markov Chains
A Markov chain is a nite-state process that can be characterized completely by
two objects the nite set of states Z = fz1 ; :; zn g and the transition matrix
= [ ij ]i;j=1;n that describes the probability of leaving state i and en