Download Work 1 - Geometric Random Variable (representing the number of

Work 1 - Geometric Random Variable (representing the number of
independent trials until the first success)
It should include,
1. Definition.
2. Example of Application.
3. Proofs for:
(a) its expected value,
(b) distribution of X + Y with X and Y i.i.d. Geom(p), p ∈ (0, 1),
(c) conditional probability of X given X + Y = n, n ∈ N with X and Y
i.i.d. Geom (p), and identify this probability function,
(d) lack-of-memory property as a characterizing property:
P (X > n + k|X > n) = P (X > k), n, k ∈ N iff X ∩ Geom(p),
p ∈ (0, 1).
Work 2 - Conditional Probability
It should include,
1. Definition of conditional probability functions (discrete and continuous
cases).
2. Deduction of probability function of Y knowing that: Y given X = n, n ∈
N with X P oisson(λ), is Binomial(n, p); identify the resulting probability
function.
3. Using conditional probability provide proofs and identify the resulting
probability function for the distribution of X1 + X2 with:
(a) Xi , independent P oisson(λi ), i = 1, 2,
(b) Xi , independent Binomial(ni , p), i = 1, 2.
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Work 3 - Moment Generating Function (mgf )
1. Definition of the moment generating function.
2. Deduce the mgf’s of the following r.v.’s: Poisson with mean λ, Binomial(n, p),
continuous Uniform and, Gamma(n, λ) and specify the Exponential case.
3. State (proofs optional but of course increase the value of the work) important properties of the mgf, including its relation with moments.
4. Obtain using mgf’s the ditribution of:
Pn
(a)
i=1 Xi with Xi independent P oisson(λi ),
Pn
(b)
i=1 Xi with Xi independent Binomial(ni , p),
Pn
(c)
i=1 Xi with Xi i.i.d. Exponential(λ).
Work 4 - Exponential Random Variable Exp(λ)
1. Definition.
2. Examples of Application.
3. Define failure rate function and obtain it the for Exp(λ). Compare the
result with the lack-of-memory property.
4. For Xi independent Exp(λi ) r.v.’s, i = 1, . . . , n, obtain:
(a) the distribution of min(X1 , . . . , Xn ) and identify the resulting distribution function,
(b) P (X1 < X2 ),
(c) P (Xj = min{X1 , . . . , Xn }).
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Work 5 - Order Statistics (random sample arranged in non-decreasing
order)
It should include,
1. Definition of order statistics (o.s.).
2. Deduction of:
(a) Joint density of the o.s. corresponding to an i.i.d. sample of continuous random variables with common density f .
(b) Distribution and density functions of X(i) .
(c) Particular cases of maxima and minima.
3. Briefly mention moments of o.s., but discuss the particular case of Uniform
random variables.
Some references:
Ross, S.M. (1983, Stochastic Processes, John Wiley & Sons, New York): for
2(a) page 66, last paragraph and, for 2(b) Exercise 2.17, page 92.
Ross, S.M. (2003, Introduction to Probability Models, 8th. edition, Academic
Press, San Diego, California): for 2(a) page 302, starting from second paragraph
and, for 2(b) Example 2.37, page 60.
Kulkarni, V.G. (1995 Modeling and Analysis of Stochastic Systems, Chapman & Hall, London): for 2 (c) particular case Uniform, formula (5.79) but not
solved. Appendix D.2 is on o.s.
Work 6 - Convergence of Sequences of Random Variables
It should include:
1. Definition of Convergence in Distribution, with examples and/or applications (e.g. CLT).
2. Definition of Convergence in Probability, with examples and/or applications.
3. Definition of Almost Sure Convergence (or convergence with probability
one).
4. Discuss the relation among the three types of convergence.
5. Discuss the Weak and Strong Laws of Large Numbers.
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Work 7 - Laplace-Stieltjes Transforms (LST)
1. Definition of LST for non-negative random variable (r.v.).
2. Properties including unicity and relation for sums of independent r.v.’s.
3. Give examples with explicit calculations including Uniform and Gamma.
4. Obtain the Gamma distribution as the sum of i.i.d. Exponential r.v.’s.
Work 8 - Random Walk
It should include,
1. Definition of Simple Random Walk.
2. Try to motivate it and describe examples of applications in different fields.
3. Describe different types of random walks and properties when appropriate.
Work 9 - Branching processes
1. Definition.
2. Describe examples and applications in different fields.
3. Describe properties including its ‘Probability of extinction’ (with at least
sketch of proof).
Work 10 - Application of Discrete Time Markov Chains to a Financial Problem
It should include,
1. Explicar o modelo markoviano usado incluindo significado das variáveis,
condições pressupostas válidas ou aproximadamente válidas, significado
das probabilidades, etc, centrado-se nas partes com ligações a matéria
dada.
2. Explicar o objectivo do modelo.
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3. Extensões e/ou outras curiosidades e/ou outras ligações com matéria dada.
Reference: Robert A. Jarrow, David Lando and Stuart M. Turnbull (1997) The
Review of Financial Studies, 10(2) 481–523.
Work 11 - Hypo- and Hyper- Exponential Random Variables
It should include,
1. Definition of both random variables that should include their construction
from exponential random variables.
2. Overview of some of their properties.
3. Examples of applications of both random variables.
4. Obtain the renewal function of a renewal process with inter-arrival times:
(a) Hyper-exponentially distributed with parameters (p, 1 − p; λ1 , λ2 ),
p ∈ (0, 1), λ1 , λ2 > 0.
(b) Hypo-exponentially distributed with parameters (λ1 , λ2 ), λ1 , λ2 > 0.
Work 12 - The M/M/1 Queueing System
1. Describe the meaning of M/M/1 in queueing theory describing the setting.
2. Describe examples of applications and usual extensions.
3. Let Ls denote the number of customers in the system, Lq the number of
customers waiting to be served, Ws the time an arriving customer spends
in the system and, Wq the time an arriving customer spends in the queue
waiting to be served. For Ls Geometrically distributed with parameter
1 − ρ, obtain (or justify when appropriate) the probability distribution of:
(a) Lq ,
(b) (Ws |Ls = k),
(c) Ws ,
(d) (Wq |Ls = k),
(e) (Wq |Wq > 0),
4. Finally, obtain the distribution function of Wq .
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