Petter Mostad
Applied Mathematics and Statistics
Chalmers and GU
MVE550 Stochastic Processes and Bayesian Inference
Re-exam August 23, 2021, 8:30 - 12:30
Examiner: Petter Mostad, phone 031-772-3579
Allowed aids: All aids are allowed.
For example you may access teaching material on any format and you may use R for computation. However, you are not allowed to communicate with any person other than the examiner and the exam guard. Total number of points: 30. To pass, at least 12 points are needed. You need to explain how you derive your answers, i.e., show the steps in computations, unless explicitly stated otherwise. There is an appendix containing information about some probability distributions.
1. (6 points) Assume p with 0 < p < 1 is a random variable with a uniform distribution.
Assume that the random variables x1, x2, . . . , are independent given p, that they can have either 0 or 1 as their value, and that given p, the probability for 1 is p.
(a) What is the marginal probability that x1 = 1?
(b) Assume that you have observed x1 = 1, x2 = 0, x3 = 0. What is the probability distribution for p given this data?
(c) Given that you have observed x1 = 1, x2 = 0, and x3 = 0, what is the probability that x4 = 0?
2. (8 points) Consider a branching process where the offspring are produced as follows: First, for k = 0, 1, 2 . . . , we add k children with a probability 2/3k+1. Then, independently, an extra child is added with a probability a, where 0 < a < 1.
(a) Compute the probability generating function for the offspring distribution. Use a form in your answer that does not contain an infinite sum.
(b) For what values of a is there a positive probability that the branching process will not go extinct?
(c) Compute the probaility of extinction for any a.
(d) Assume you have observed the offspring process above in N cases, and that in r of these cases, the number of offspring was zero. Assuming a uniform prior for a and given this information, write down a function of a that is proportional to the posterior density for a on the interval [0, 1].
3. (8 points) Two types of particles arrive at a detector according to independent Poisson processes: Type alpha arrives at a rate of 2.1 per second and type beta at a rate of 4.9 per second. You may answer the questions below with computed numbers, or for example with explicit R code for how to compute the result. Always explain how you reach your result.
(a) What is the probability that exactly 4 particles reach the detector in the interval between 4.0 and 4.5 seconds from the start of detections?
(b) Given that 5 particles reach the detector within the first second of detections, what is the probability that exactly 3 of these are alpha particles?
(c) Given that 7 beta particles arrive in the first 2 seconds, what is the probability that none of these beta particles arrive during the last half of this period?
(d) Given that 6 alpha particles arrive during the first two seconds, what is the probability that the 6th beta particle arrives in the interval [1.7, 2] seconds?
(e) What is the probability that the first particle to arrive in the detector is an alpha particle?
4. (4 points) A celltype can exist in four states: A growth state, a resting state, a reproduction state, or it can be dead. It starts in the growth state. From the growth state it moves to the resting state with a rate of 0.3 and to the reproduction state with a rate 0.1, or it dies with a rate of 0.1. From the resting state it moves to the growth state with a rate of 0.2, or it dies with a rate of 0.05. From the reproduction state it moves to the resting state at a rate 0.5, or it dies at a rate of 0.3.
(a) What is the expected life time of the cell? Compute the answer numerically, or describe in detail how such computations are done.
(b) Write down the transition matrix for the corresponding discrete-time embedded Markov chain.
5. (4 points) The future price of a particular patent is modelled using a Brownian motion Bt with a drift: The price after t years is modelled as V(t) = 8000 + 400t + 500Bt
(a) What is the probability that the price is above 10000 after 2 years?
(b) What is the expected time at which the price reaches 10000?
Appendix: Some probability distributions
The Bernoulli distribution
If x ∈{0, 1} has a Bernoulli distribution with parameter 0 ≤p ≤1, then the probability mass function is π(x) = px(1 −p)1−x.
We write x | p ∼Bernoulli(p) and π(x | p) = Bernoulli(x; p).
The Beta distribution
If x ∈[0, 1] has a Beta distribution with parameters with α > 0 and β > 0 then the density is π(x | α, β) = Γ(α + β) Γ(α)Γ(β) xα−1(1 −x)β−1.
We write x | α, β ∼Beta(α, β) and π(x | α, β) = Beta(x; α, β).
The Beta-Binomial distribution
If x ∈{0, 1, 2, . . . , n} has a Beta-Binomial distribution, with n a positive integer and parameters α > 0 and β > 0, then the probability mass function is π(x | n, α, β) = n x !Γ(x + α)Γ(n −x + β)Γ(α + β) Γ(α)Γ(β)Γ(n + α + β).
We write x | n, α, β ∼Beta-Binomial(n, α, β) and π(x | n, α, β) = Beta-Binomial(x; n, α, β).
The Binomial distribution
If x ∈{0, 1, 2, . . . , n} has a Binomial distribution, with n a positive integer and 0 ≤p ≤1, then the probability mass function is π(x | n, p) = n x ! px(1 −p)n−x.
We write x | n, p ∼Binomial(n, p) and π(x | n, p) = Binomial(x; n, p).
The Dirichlet distribution
If x = (x1, x2, . . . , xn) has a Dirichlet distribution, with xi ≥0 and Pn i=1 xi = 1 and with parameters α = (α1, . . . , αn) with α1 > 0, ..., αn > 0, then the density function is π(x | α) = Γ(α1 + α2 + · · · + αn) Γ(α1)Γ(α2) · · · Γ(αn) pα1−1 1 pα2−1 2 · · · pαn−1 n.
We write x | α ∼Dirichlet(α) and π(x | α) = Dirichlet(x; α).
The Exponential distribution
If x ≥0 has an Exponential distribution with parameter λ > 0, then the density is π(x | λ) = λ exp(−λx)
We write x | λ ∼Exponential(λ) and π(x | λ) = Exponential(x; λ). The expectation is 1/λ and the variance is 1/λ2.
The Gamma distribution
If x > 0 has a Gamma distribution with parameters α > 0 and β > 0 then the density is π(x | αβ) = βα Γ(α)xα−1 exp(−βx).
We write x | α, β ∼Gamma(α, β) and π(x | α, β) = Gamma(x; α, β).
The Geometric distribution
If x ∈{1, 2, 3, . . . } has a Geometric distribution with parameter p ∈(0, 1), the probability mass function is π(x | p) = p(1 −p)x−1
We write x | p ∼Geometric(p) and π(x | p) = Geometric(x; p). The expectation is 1/p and the variance (1 −p)/p2.
The Normal distribution
If the real x has a Normal distribution with parameters µ and σ2, its density is given by π(x | µ, σ2) = 1 √ 2πσ2 exp( −1 2σ2(x −µ)2 ).
We write x | µ, σ2 ∼Normal(µ, σ2) and π(x | µ, σ2) = Normal(x; µ, σ2).
