Questions and Answers of Probability And Stochastic Modeling

Let $Y_{0}, Y_{1}, \ldots$ be a sequence of independent random variables, which are identically distributed as $N(0,1)$. Are the stochastic sequences $\left\{X_{0}, X_{1}, \ldots\right\}$
Let $Y_{0}, Y_{1}, \ldots$ be a sequence of independent random variables with finite mean values. Show that the discrete-time stochastic process $\left\{X_{0}, X_{1}, \ldots\right\}$ generated
Let a discrete-time stochastic process $\left\{X_{0}, X_{1}, \ldots\right\}$ be defined by\[X_{n}=Y_{0} \cdot Y_{1} \cdots Y_{n}\]where the random variables $Y_{i}$ are independent and have a
Determine the mean value of the loss immediately before the win when applying the doubling strategy, i.e., determine $E\left(X_{N-1}\right)$ (example 10.6).Data from Example 10.6 Example 10.6 The
Why is theorem 10.2 not applicable to the sequence of 'winnings' $\left\{X_{1}, X_{2}, \ldots\right\}$, which arises by applying the doubling strategy (example 10.6)?Data from Example 10.6Data from
Jean is not happy with the winnings he can make when applying the 'doubling strategy'. Hence, under otherwise the same assumptions and notations as in example 10.6, he triples his bet size after
Starting at value 0 , the profit of an investor increases per week by $\$ 1$ with probbability $p, p>1 / 2$, or decreases per week by one unit with probability $1-p$. The weekly increments of
Starting at value 0 , the fortune of an investor increases per week by $\$ 200$ with probability $3 / 8$, remains constant with probability $3 / 8$, and decreases by $\$ 200$ with probability
Let $X_{0}$ be uniformly distributed over $[0, T], X_{1}$ be uniformly distributed over $\left[0, X_{0}\right]$, and, generally, $X_{i+1}$ be uniformly distributed over \(\left[0,
Let $\left\{X_{1}, X_{2}, \ldots\right\}$ be a homogeneous discrete-time Markov chain with state space $\mathbf{Z}=\{0,1, \ldots, n\}$ and transition probabilities\[p_{i j}=P\left(X_{k+1}=j \mid
Show that if $L$ is a stopping time for a stochastic process with discrete or continuous time and \(0
Let $\{N(t), t \geq 0\}$ be a nonhomogeneous Poisson process with intensity function $\lambda(t)$ and trend function\[\Lambda(t)=\int_{0}^{t} \lambda(x) d x\]Check whether the stochastic process
Show that every stochastic process $\{X(t), t \in \mathbf{T}\}$ satisfying\[E(|X(t)|)
Verify that the probability density $f_{t}(x)$ of $B(t)$,\[f_{t}(x)=\frac{1}{\sqrt{2 \pi t} \sigma} e^{-x^{2} /\left(2 \sigma^{2} t\right)}, \quad t>0\]satisfies with a positive constant $c$
Determine the conditional probability density of $B(t)$ given \(B(s)=y, 0 \leq s
Prove that the stochastic process $\{\bar{B}(t), 0 \leq t \leq 1\}$ given by $\bar{B}(t)=B(t)-t B(1)$ is the Brownian bridge.
Let $\{\bar{B}(t), 0 \leq t \leq 1\}$ be the Brownian bridge. Prove that the stochastic process\[\{S(t), t \geq 0\} \text { defined by } S(t)=(t+1) \bar{B}\left(\frac{t}{t+1}\right)\]is the
Determine the probability density of \(B(s)+B(t), 0 \leq s
Let $n$ be any positive integer. Determine mean value and variance of\[X(n)=B(1)+B(2)+\cdots+B(n)\]
Check whether for any positve $\tau$ the stochastic process $\{V(t), t \geq 0\}$ defined by\[V(t)=B(t+\tau)-B(t)\]is weakly stationary.
Let $X(t)=S^{3}(t)-3 t S(t)$. Prove that $\{X(t), t \geq 0\}$ is a continuous-time martingale, i.e., show that\[E(X(t) \mid X(y), y \leq s)=X(s), \quad s
Show by a counterexample that the Ornstein-Uhlenbeck process does not have independent increments.
(1) What is the mean value of the first passage time of the reflected Brownian motion $\{|B(t)|, t \geq 0\}$ with regard to a positive level $x$ ?(2) Determine the distribution function of
Starting from $x=0$, a particle makes independent jumps of length\[\Delta x=\sigma \sqrt{\Delta t}\]to the right or to the left every $\Delta t$ time units. The respective probabilities of jumps
Let $\{D(t), t \geq 0\}$ be a Brownian motion with drift with paramters $\mu$ and $\sigma$. Determine $E\left(\int_{0}^{t}(D(s))^{2} d s\right)$.
Show that for $c>0$ and $d>0$\[P(B(t) \leq c t+d \text { for all } t \geq 0)=1-e^{-2 c d / \sigma^{2}}\]
At time $t=0$ a speculator acquires an American call option with infinite expiration time and strike price $x_{s}$. The price [in \$] of the underlying risky security at time $t$ is given by
The price of a unit of a share at time point $t$ is $X(t)=10 e^{D(t)}, t \geq 0$, where $\{D(t), t \geq 0\}$ is a Brownian motion process with drift parameter $\mu=-0.01$ and volatility
The value (in $\$$ ) of a share per unit develops, apart from the constant factor 10 , according to a geometric Brownian motion $\{X(t), t \geq 0\}$ given by\[X(t)=10 e^{B(t)}, 0 \leq t \leq
The value of a share per unit develops according to a geometric Brownian motion with drift given by\[X(t)=10 e^{0.2 t+0.1 S(t)}, t \geq 0\]where $\{S(t), t \geq 0\}$ is the standardized Brownian
The random price $X(t)$ of a risky security per unit at time $t$ is\[X(t)=5 e^{-0.01 t+B(t)+0.2|B(t)|}\]where $\{B(t), t \geq 0\}$ is the Brownian motion with volatility\[\sigma=0.04\]At time
At time $t=0$ a speculator acquires a European call option with strike price $x_{S}$ and finite expiration time $\tau$. Thus, the option can only be exercised at time $\tau$ at price
Show that\[E\left(e^{\alpha U(t)}\right)=e^{\alpha^{2} t^{3} / 6}\]for any constant $\alpha$, where $U(t)$ is the integrated standard Brownian motion:\[U(t)=\int_{0}^{t} S(x) d x, t \geq 0\]
For any fixed positive $\tau$, let the stochastic process $\{V(t), t \geq 0\}$ be given by\[V(t)=\int_{t}^{t+\tau} S(x) d x\]Is $\{V(t), t \geq 0\}$ weakly stationary?
Let $\{X(t), t \geq 0\}$ be the cumulative repair cost process of a system with\[X(t)=0.01 e^{D(t)}\]where $\{D(t), t \geq 0\}$ is a Brownian motion with drift and parameters\[\mu=0.02 \text {
Define the stochastic process $\{X(t), t \in \mathbf{R}\}$ by\[X(t)=A \cos (\omega t+\Phi)\]where $A$ and $\Phi$ are independent random variables with $E(A)=0$ and $\Phi$ is uniformly
A weakly stationary, continuous-time process has covariance function\[C(\tau)=\sigma^{2} e^{-\alpha|\tau|}\left(\cos \beta \tau-\frac{\alpha}{\beta} \sin \beta|\tau|\right)\]Prove that its spectral
A weakly stationary continuous-time process has covariance function\[C(\tau)=\sigma^{2} e^{-\alpha|\tau|}\left(\cos \beta \tau+\frac{\alpha}{\beta} \sin \beta|\tau|\right)\]Prove that its spectral
A weakly stationary continuous-time process has covariance function\[C(\tau)=a^{-b \tau^{2}} \text { for } a>0, b>0\]Prove that its spectral density is given by\[s(\omega)=\frac{a}{2 \sqrt{\pi b}}
Define a weakly stationary stochastic process $\{V(t), t \geq 0\}$ by\[V(t)=S(t+1)-S(t)\]where $\{S(t), t \geq 0\}$ is the standard Brownian motion process.Prove that its spectral density is
A weakly stationary, continuous-time stochastic process has spectral density\[s(\omega)=\sum_{k=1}^{n} \frac{\alpha_{k}}{\omega^{2}+\beta_{k}^{2}}, \quad \alpha_{k}>0\]Prove that its covariance
A weakly stationary, continuous-time stochastic process has spectral density\[s(\omega)= \begin{cases}0 \text { for }|\omega|2 \omega_{0}, \\ a^{2} \text { for } & \omega_{0} \leq|\omega| \leq 2
Let $\mathbf{Z}=\{0,1\}$ be the state space and\[\mathbf{P}(t)=\left(\begin{array}{cc} e^{-t} & 1-e^{-t} \\ 1-e^{-t} & e^{-t} \end{array}\right)\]the transition matrix of a continuous-time
A system fails after a random lifetime $L$. Then it waits a random time $W$ for renewal. A renewal takes another random time $Z$. The random variables $L, W$, and $Z$ have exponential
Consider a 1 -out-of- 2 system, i.e., the system is operating when at least one of its two subsystems is operating. When a subsystem fails, the other one continues to work. On its failure, the joint
A copy center has 10 copy machines of the same type which are in constant use. The times between two successive failures of a machine have an exponential distribution with mean value 100 hours. There
Consider the two-unit system with standby redundancy discussed in example 9.5 a) on condition that the lifetimes of the units are exponential with respective parameters $\lambda_{1}$ and
Consider the two-unit system with parallel redundancy discussed in example 9.6 on condition that the lifetimes of the units are exponential with parameters $\lambda_{1}$ and $\lambda_{2}$,
The system considered in example 9.7 is generalized as follows: If the system makes a direct transition from state 0 to the blocking state 2 , then the subsequent renewal time is exponential with
Consider a two-unit system with standby redundancy and one mechanic. All repair times of failed units have an Erlang distribution with parameters $n=2$ and $\mu$. Apart from this, the other model
Consider a two-unit parallel system (i.e., the system operates if at least one unit is operating). The lifetimes of the units have an exponential distributions with parameter $\lambda$. There is
When being in states 0,1 , and 2 , a (pure) birth process $\{X(t), t \geq 0\}$ with state space $\mathbf{Z}=\{0,1,2, \ldots\}$ has the respective birth rates\[\lambda_{0}=2, \lambda_{1}=3,
Consider a linear birth process with state space $\mathbf{Z}=\{0,1,2, \ldots\}$ and transition rates $\lambda_{j}=j \lambda, j=0,1, \ldots$(1) Given $X(0)=1$, determine the distribution
The number of physical particles of a particular type in a closed container evolves as follows: There is one particle at time $t=0$. Its splits into two particles of the same type after an
A death process with state space $\mathbf{Z}=\{0,1,2, \ldots\}$ has death rates\[\mu_{0}=0, \mu_{1}=2, \text { and } \mu_{2}=\mu_{3}=1\]Given $X(0)=3$, determine $p_{j}(t)=P(X(t)=j)$ for
A linear death process $\{X(t), t \geq 0\}$ has death rates $\mu_{j}=j \mu ; j=0,1, \ldots$.(1) Given $X(0)=2$, determine the distribution function of the time to entering state 0 ('lifetime'
At time $t=0$ there are an infinite number of molecules of type $a$ and $2 n$ molecules of type $b$ in a two-component gas mixture. After an exponential random time with parameter $\mu$ any
At time $t=0$ a cable consists of 5 identical, intact wires. The cable is subject to a constant load of $100 \mathrm{kp}$ such that in the beginning each wire bears a load of $20 \mathrm{kp}$.
Let $\{X(t), t \geq 0\}$ be a death process with $X(0)=n$ and positive death rates $\mu_{1}, \mu_{2}, \ldots, \mu_{n}$.Prove: If $Y$ is an exponential random variable with parameter
A birth- and death process has state space $\mathbf{Z}=\{0,1, \ldots, n\}$ and transition rates\[\lambda_{j}=(n-j) \lambda \text { and } \mu_{j}=j \mu ; j=0,1, \ldots, n\]Determine its stationary
Check whether or under what restrictions a birth- and death process with transition rates\[\lambda_{j}=\frac{j+1}{j+2} \lambda \text { and } \mu_{j}=\mu ; j=0,1, \ldots\]has a stationary state
A birth- and death process has transition rates\[\lambda_{j}=(j+1) \lambda \text { and } \mu_{j}=j^{2} \mu ; j=0,1, \ldots ; 0
Consider the following deterministic models for the mean (average) development of the size of populations:(1) Let $m(t)$ be the mean number of individuals of a population at time $t$. It is
A computer is connected to three terminals (for example, measuring devices). It can simultaneously evaluate data records from only two terminals. When the computer is processing two data records and
Under otherwise the same assumptions as in exercise 9.22, it is assumed that a data record, which has been waiting in the buffer a random patience time, will be deleted as being no longer up to date.
Under otherwise the same assumptions as in exercise 9.22 , it is assumed that a data record will be deleted when its total sojourn time in the buffer and computer exceeds a random time $Z$, where
A small filling station in a rural area provides diesel for agricultural machines. It has one diesel pump and waiting capacity for 5 machines. On average, 8 machines per hour arrive for diesel. An
Consider a two-server loss system. Customers arrive according to a homogeneous Poisson process with intensity $\lambda$. A customer is always served by server 1 when this server is idle, i.e., an
A two-server loss system is subject to a homogeneous Poisson input with intensity $\lambda$. The situation considered in exercise 9.26 is generalized as follows: If both servers are idle, a
A single-server waiting system is subject to a homogeneous Poisson input with intensity $\lambda=30\left[h^{-1}\right]$. If there are not more than 3 customers in the system, the service times have
Taxis and customers arrive at a taxi rank in accordance with two independent homogeneous Poisson processes with intensities\[\lambda_{1}=4\left[h^{-1}\right] \text { and }
A transport company has 4 trucks of the same type. There are 2 maintenance teams for repairing the trucks after a failure. Each team can repair only one truck at a time and each failed truck is
Ferry boats and customers arrive at a ferry station in accordance with two independent homogeneous Poisson processes with intensities $\lambda$ and $\mu$, respectively. If there are $k$
The life cycle of an organism is controlled by shocks (e.g., accidents, virus attacks) in the following way: A healthy organism has an exponential lifetime $L$ with parameter $\lambda_{h}$. If a
Customers arrive at a waiting system of type $M / M / 1 / \infty$ with intensity $\lambda$. As long as there are less than $n$ customers in the system, the server remains idle. As soon as the
At time $t=0$ a computer system consists of $n$ operating computers. As soon as a computer fails, it is separated from the system by an automatic switching device with probability $1-p$. If a
A waiting-loss system of type $M / M / 1 / 2$ is subject to two independent Poisson inputs 1 and 2 with respective intensities $\lambda_{1}$ and $\lambda_{2}$, which are referred to as type
A queueing network consists of two servers 1 and 2 in series. Server 1 is subject to a homogeneous Poisson input with intensity $\lambda=5$ an hour. A customer is lost if server 1 is busy. From
A queueing network consists of three nodes (queueing systems) 1,2 , and 3 , each of type $M / M / 1 / \infty$. The external inputs into the nodes have respective intensities\[\lambda_{1}=4,
A closed queueing network consists of 3 nodes. Each one has 2 servers. There are 2 customers in the network. After having been served at a node, a customer goes to one of the others with equal
Depending on demand, a conveyor belt operates at 3 different speed levels 1,2 , and 3. A transition from level $i$ to level $j$ is made with probability $p_{i j}$ with\[p_{12}=0.8, p_{13}=0.2,
The mean lifetime of a system is 620 hours. There are two failure types: Repairing the system after a type 1 -failure requires 20 hours on average and after a type 2 -failure 40 hours on average.
Under otherwise the same model assumptions as in example 9.25, determine the stationary probabilities of the states 0,1 , and 2 introduced there on condition that the service time $B$ is a constant
A system has two different failure types: type 1 and type 2 . After a type $i$-failure the system is said to be in failure state $i ; i=1,2$. The time $L_{i}$ to a type $i$-failure has an
The occurrence of catastrophic accidents at Sosal & Sons follows a homogeneous Poisson process with intensity $\lambda=3$ a year.(1) What is the probability $p_{\geq 2}$ that at least two
By making use of the independence and homogeneity of the increments of a homogeneous Poisson process with intensity $\lambda$, show that its covariance function is given by\[C(s, t)=\lambda \min
The number of cars which pass a certain intersection daily between $12: 00$ and 14:00 follows a homogeneous Poisson process with intensity $\lambda=40$ per hour. Among these there are $2.2 \%$
A Geiger counter is struck by radioactive particles according to a homogeneous Poisson process with intensity $\lambda=1$ per 12 seconds. On average, the Geiger counter only records 4 out of 5
The location of trees in an even, rectangular forest stand of size $200 \mathrm{~m} \times 500 \mathrm{~m}$ follows a homogeneous Poisson distribution with intensity $\lambda=1$ per \(25
An electronic system is subject to two types of shocks, which occur independently of each other according to homogeneous Poisson processes with intensities\[\lambda_{1}=0.002 \text { and }
A system is subjected to shocks of types 1,2 , and 3 , which are generated by independent homogeneous Poisson processes with respective intensities per hour $\lambda_{1}=0.2, \lambda_{2}=0.3$, and
Claims arrive at a branch of an insurance company according a homogeneous Poisson process with an intensity of $\lambda=0.4$ per working hour. The claim size $Z$ has an exponential distribution
Consider two independent homogeneous Poisson processes 1 and 2 with respective intensities $\lambda_{1}$ and $\lambda_{2}$. Determine the mean value of the random number of events of process 2 ,
Let $\{N(t), t \geq 0\}$ be a homogeneous Poisson process with intensity $\lambda$.Prove that for an arbitrary, but fixed, positive $h$ the stochastic process $(X(t), t \geq 0\}$ defined by
Let a homogeneous Poisson process have intensity $\lambda$, and let $T_{i}$ be the time point at which the $i$ th Poisson event occurs. For $t \rightarrow \infty$, determine and sketch the
Statistical evaluation of a large sample justifies to model the number of cars which arrive daily for petrol between 0:00 and 4:00 a.m. at a particular filling station by an inhomogeneous Poisson
Let $\{N(t), t \geq 0\}$ be an inhomogeneous Poisson process with intensity function\[\lambda(t)=0.8+2 t, \quad t \geq 0\]Determine the probability that at least 500 Poisson events occur in
Let $\{N(t), t \geq 0\}$ be a nonhomogeneous Poisson process with trend function $\Lambda(t)$ and arrival time point $T_{i}$ of the $i$ th Poisson event.Given $N(t)=n$, show that the random
Clients arrive at an insurance company according to a mixed Poisson process the structure parameter $L$ of which has a uniform distribution over the interval $[0,1]$.(1) Determine the state
A system is subjected to shocks of type 1 and type 2 , which are generated by independent Pólya processes $\left\{N_{L_{1}}(t), t \geq 0\right\}$ and $\left\{N_{L_{2}}(t), t \geq 0\right\}$ with

Showing 1 - 100 of 1215