Questions and Answers of Probability And Stochastic Modeling

Prove the multinomial criterion (formula 7.55, page 280).Data from formula 7.55 P(NL(tk-1, tk)=ik; k = 1,2,...,n|N(tn) = 1) i! 11!12!in! ()" 71 12-11 12 tn in tn
An insurance company has a premium income of $\$ 106080$ per day. The claim sizes are iid random variables and have an exponential distribution with variance $4 \cdot 10^{6}\left[\$^{2}\right]$.
Pramod is setting up an insurance policy for low-class cars (homogeneous portfolio) over an infinite time horizon. Based on previous statistical work, he expects that claims will arrive according to
The lifetime $L$ of a system has a Weibull-distribution with distribution function\[F(t)=P(L \leq t)=1-e^{-0.1 t^{3}}, t \geq 0\](1) Determine its failure rate $\lambda(t)$ and its integrated
A system is maintained according to Policy 3 over an infinite time span. It has the same lifetime distribution and minimal repair cost parameter as in exercise 7.20. As with exercise 7.20, let
A system starts working at time $t=0$. Its lifetime has approximately a normal distribution with mean value $\mu=125$ hours and standard deviation $\sigma=40$ hours. After a failure, the system
(1) Use the Laplace transformation to find the renewal function $H(t)$ of an ordinary renewal process whose cycle lengths have an Erlang distribution with parameters $n=2$ and $\lambda$.(2) For
An ordinary renewal function has the renewal function $H(t)=t / 10$. Determine the probability $P(N(10) \geq 2)$.
A system is preventively replaced by an identical new one at time points $\tau, 2 \tau, \ldots$ If failures happen in between, then the failed system is replaced by an identical new one as well.
Given the existence of the first three moments of the cycle length $Y$ of an ordinary renewal process, verify the formulas (7.112).Data from 7.112 11+02 E(S)= and E(S2) 113 2 3
(1) Verify that the probability $p(t)=P(N(t)$ is odd) satisfies\[p(t)=F(t)-\int_{0}^{t} p(t-x) f(x) d x, \quad f(x)=F^{\prime}(x)\](2) Determine this probability if the cycle lengths are
Verify that the second moment of $N(t)$, denoted as $H_{2}(t)=E\left(N^{2}(t)\right)$, satisfies the integral equation\[H_{2}(t)=2 H(t)-F(t)+\int_{0}^{t} H_{2}(t-x) f(x) d x .\]Verify the
The times between the arrivals of successive particles at a counter generate an ordinary renewal process. Its random cycle length $Y$ has distribution function $F(t)$ and mean value $\mu=E(Y)$.
The cycle length distribution of an ordinary renewal process is given by the distribution function $F(t)=1-e^{-t^{2}}, t \geq 0$ (Rayleigh distribution).(1) What is the statement of theorem 7.13 if
Let be $A(t)$ the forward and $B(t)$ the backward recurrence times of an ordinary renewal process at time $t$. For $x>y / 2$, determine functional relationships between $F(t)$ and the
Let $(Y, Z)$ be the typical cycle of an alternating renewal process, where $Y$ and $Z$ have an Erlang distribution with joint parameter $\lambda$ and parameters $n=2$ and $n=1$,
The time intervals between successive repairs of a system generate an ordinary renewal process $\left\{Y_{1}, Y_{2}, \ldots\right\}$ with typical cycle length $Y$. The costs of repairs are
(1) Determine the ruin probability $p(x)$ of an insurance company with an initial capital of $x=\$ 20000$ and operating parameters\[1 / \mu=2\left[h^{-1}\right], v=\$ 800 \text { and }
Under otherwise the same assumptions as made in example 7.10, determine the ruin probability if the random claim size $M$ has density\[b(y)=\lambda^{2} y e^{-\lambda y}, \lambda>0, y \geq
Claims arrive at an insurance company according to an ordinary renewal process $\left\{Y_{1}, Y_{2}, \ldots\right\}$. The corresponding claim sizes $M_{1}, M_{2}, \ldots$ are independent and
A Markov chain $\left\{X_{0}, X_{1}, \ldots\right\}$ has state space $\mathbf{Z}=\{0,1,2\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{ccc} 0.5 & 0 & 0.5 \\ 0.4 & 0.2 & 0.4 \\ 0 &
A Markov chain $\left\{X_{0}, X_{1}, \ldots\right\}$ has state space $\mathbf{Z}=\{0,1,2\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{ccc} 0.2 & 0.3 & 0.5 \\ 0.8 & 0.2 & 0 \\ 0.6 &
A Markov chain $\left\{X_{0}, X_{1}, \ldots\right\}$ has state space $\mathbf{Z}=\{0,1,2\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{ccc} 0 & 0.4 & 0.6 \\ 0.8 & 0 & 0.2 \\ 0.5 &
Let $\left\{Y_{0}, Y_{1}, \ldots\right\}$ be a sequence of independent, identically distributed binary random variables with $P\left(Y_{i}=0\right)=P\left(Y_{i}=1\right)=1 / 2 ; i=0,1, \ldots$.
A Markov chain $\left\{X_{0}, X_{1}, \ldots\right\}$ has state space $\mathbf{Z}=\{0,1,2,3\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{llll} 0.1 & 0.2 & 0.4 & 0.3 \\ 0.2 & 0.3 &
Let $\left\{X_{0}, X_{1}, \ldots\right\}$ be an irreducible Markov chain with state space $\mathbf{Z}=\{1,2, \ldots, n\}$, \(n
Prove formulas (8.20), page 346 , for the mean times to absorption in a random walk with two absorbing barriers (example 8.3).Data from 8.20Data from Example 8.3 1 1-(q/p)" m(n): p-q 1-(q/p). m(n)
Show that the vector $\pi=\left(\pi_{1}=\alpha, \pi_{2}=\beta, \pi_{3}=\gamma\right)$, determined in example 8.6 , is a stationary initial distribution with regard to a Markov chain which has the
A source emits symbols 0 and 1 for transmission to a receiver. Random noises $S_{1}, S_{2}, \ldots$ successively and independently affect the transmission process of a symbol in the following way:
Weather is classified as (predominantly) sunny (S) and (predominantly) cloudy (C), where $\mathrm{C}$ includes rain. For the town of Musi, a fairly reliable prediction of tomorrow's weather can
A supplier of toner cartridges of a certain brand checks her stock every Monday. If the stock is less than or equal to $s$ cartridges, she orders an amount of $S-s$ cartridges, which will be
A Markov chain has state space $\mathbf{Z}=\{0,1,2,3,4\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{ccccc} 0.5 & 0.1 & 0.4 & 0 & 0 \\ 0.8 & 0.2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0
A Markov chain has state space $\mathbf{Z}=\{0,1,2,3\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0.4 & 0.6 & 0 & 0 \\ 0.1 & 0.4 & 0.2 & 0.3
A Markov chain has state space $\mathbf{Z}=\{0,1,2,3,4\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{ccccc} 0 & 0.2 & 0.8 & 0 & 0 \\ 0 & 0 & 0 & 0.9 & 0.1 \\ 0 & 0 & 0 & 0.1 & 0.9 \\
A Markov chain has state space $\mathbf{Z}=\{0,1,2,3,4\}$ and transition matrix\[\mathbf{P}=\left(\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0.2 & 0.2 & 0.2 & 0.4 & 0 \\ 0.2
Determine the stationary distribution of the random walk considered in example 8.12 on condition \(p_{i}=p, 0Data from Example 8.12 Example 8.12 A particle jumps from x = i to x=0 with probability p;
The weekly power consumption of a town depends on the weekly average temperature in that town. The weekly average temperature, observed over a long time span in the month of August, has been
A household insurer knows that the total annual claim size $X$ of clients in a certain portfolio hasy a normal distribution with mean value $\$ 800$ and standard deviation $\$ 260$. The insurer
Two gamblers 1 and 2 begin a game with stakes of sizes $\$ 3$ and $\$ 4$, respectively. After each move a gambler either wins or loses $\$ 1$ or the gambler's stake remains constant. These
Analogously to example 8.17 (page 369), consider a population with a maximal size of $z=5$ individuals, which comprises at the beginning of its observation 3 individuals. Its birth and death
Let the transition probabilities of a birth and death process be given by\[p_{i}=\frac{1}{1+[i /(i+1)]^{2}} \text { and } q_{i}=1-p_{i} ; i=1,2, \ldots ; p_{0}=1\]Show that the process is transient.
Let $i$ and $j$ be two different states with $f_{i j}=f_{j i}=1$. Show that both $i$ and $j$ are recurrent.
The respective transition probabilities of two irreducible Markov chains 1 and 2 with common state space $\mathbf{Z}=\{0,1, \ldots\}$ are for all $i=0,1, \ldots$,(1)\(p_{i+1}=\frac{1}{i+2}, \quad
Let $N_{i}$ be the random number of time periods a discrete-time Markov chain stays in state $i$ (sojourn time of the Markov chain in state $i$ ).Determine $E\left(N_{i}\right)$ and
A Galton-Watson process starts with one individual. The random number of offspring $Y$ of this individual has the $z$-transform\[M(z)=(0.6 z+0.4)^{3}\](1) What type of probability distribution
A Galton-Watson process starts with one individual. The random number of offspring $Y$ of this individual has the $z$-transform\[M(z)=e^{1.5(z-1)}\](1) What is the underlying probability
(1) Determine the $z$-transform of the truncated, $p_{0}$ - modified geometric distribution given by formula (8.62).(2) Determine the corresponding probability of extinction $\pi_{0}$ if\[m=6,
Assume a Galton-Watson process starts with $X_{0}=n>1$ offspring.Determine the corresponding probability of extinction given that the same GaltonWatson process, when starting with one offspring,
Given $X_{0}=1$, show that the probability of extinction $\pi_{0}$ satisfies equation\[M\left(\pi_{0}\right)=\pi_{0}\]by applying the total probability rule (condition with regard to the number
In a game reserve, the random position $(X, Y)$ of a leopard has a uniform distribution in a semicircle with radius $r=10 \mathrm{~km}$ (figure). Determine $E(X)$ and $E(Y)$. V Y 10 -10 0 X
From a circle with radius $R=9$ and center $(0,0)$ a point is randomly selected.(1) Determine the mean value of the distance of this point to the nearest point at the periphery of the circle.(2)
$X$ and $Y$ are independent, exponentially with parameter $\lambda=1$ distributed random variables. Determine(1) $E(X-Y)$,(2) $E(|X-Y|)$, and(3) distribution function and density of
$X$ and $Y$ are independent random variables with$E(X)=E(Y)=5, \operatorname{Var}(X)=\operatorname{Var} Y)=9$, and let $U=2 X+3 Y$ and $V=3 X-2 Y$.Determine \(E(U), E(V),
$X$ and $Y$ are independent, in the interval $[0,1]$ uniformly distributed random variables. Determine the densities of(1) $Z=\min (X, Y)$, and (2) $Z=X Y$.
$X$ and $Y$ are independent and $N(0,1)$-distributed. Determine the density $f_{Z}(z)$ of\[Z=X / Y\]Which type of probability distributions does $f_{Z}(z)$ belong to?
$X$ and $Y$ are independent and identically Cauchy distributed with parameters $\lambda=1$ and $\mu=0$, i.e. they have densities\[f_{X}(x)=\frac{1}{\pi} \frac{1}{1+x^{2}}, \quad
The joint density of the random vector $(X, Y)$ is\[f(x, y)=6 x^{2} y, \quad 0 \leq x, y \leq 1\]Determine the distribution density of the product $Z=X Y$.
The random vector $(X, Y)$ has the joint density\[f_{X, Y}(x, y)=2 e^{-(x+y)} \text { for } 0 \leq x \leq y
The resistance values $X, Y$, and $Z$ of 3 resistors connected in series are assumed to be independent, normally distributed random variables with respective mean values 200,300 , and
A supermarket employs 24 shopassistants. 20 of them achieve an average daily turnover of $\$ 8000$, whereas 4 achieve an average daily turnover of $\$ 10000$. The corresponding standard
A helicopter is allowed to carry at most 8 persons given that their total weight does not exceed $620 \mathrm{~kg}$. The weights of the passengers are independent, identically normally distributed
Let $X$ be the height of the woman and $Y$ be the height of the man in married couples in a certain geographical region. By analyzing a sufficiently large sample, a statistician found that the
A target, which is located at point $(0,0)$ of the $(x, y)$ - coordinate system, is subject to permanent shellfire. The random coordinates $X$ and $Y$ of the hitting point of a shell are
On average, $6 \%$ of the citizens of a large town suffer from severe hypertension. Let $X$ be the number of people in a sample of $n$ randomly selected citizens from this town which suffer
The measurement error $X$ of a measuring device has mean value $E(X)=0$ and variance $\operatorname{Var}(X)=0.16$. The random outcomes of $n$ independent measurements are \(X_{1}, X_{2},
A manufacturer of $T V$ sets knows from past experience that $4 \%$ of his products do not pass the final quality check.(1) What is the probability that in the total monthly production of 2000
The daily demand for a certain medication in a country is given by a random variable $X$ with mean value 28 packets per day and with a variance of 64 . The daily demands are independent of each
According to the order, the rated nominal capacitance of condensers in a large delivery should be $300 \mu F$. Their actual rated nominal capacitances are, however, random variables $X$
A digital transmission channel distorts on average 1 out of 10000 bits during transmission. The bits are transmitted independently of each other.(1) Give the exact formula for the probability of the
Solve the problem of example 2.4 (page 51) by making use of the normal approximation to the binomial distribution and compare with the exact result.Data from Example 2.4A power station supplies power
Solve the problem of example 2.6 (page 54) by making use of the normal approximation to the hypergeometric distribution and compare with the exact result.Data from Example 2.6A customer knows that on
The random number of asbestos particles per $1 \mathrm{~mm}^{3}$ in the dust of an industrial area is Poisson distributed with parameter $\lambda=8$.What is the probability that in \(1
The number of e-mails, which daily arrive at a large company, is Poisson distributed with parameter\[\lambda=22400\]What is the probability that daily between between 22300 and 22500 e-mails arrive?
In $1 \mathrm{~kg}$ of a tapping of cast iron melt there are on average 1.2 impurities.What is the probability that in a $1000 \mathrm{~kg}$ tapping there are at least 1400 impurities?The spacial
After six weeks, 24 seedlings, which had been planted at the same time, reach the random heights $X_{1}, X_{2}, \ldots, X_{24}$, which are independent, identically exponentially distributed as
Under otherwise the same assumptions as in exercise 5.12, only 6 seedlings had been planted. Determine(1) the exact probability that the arithmetic mean\[\bar{X}_{6}=\frac{1}{6} \sum_{i=1}^{6}
The continuous random variable $X$ is uniformly distributed on $[0,2]$.(1) Draw the graph of the function\[p(\varepsilon)=P(|X-1| \geq \varepsilon)\]in dependence of \(\varepsilon, 0 \leq
A stochastic process $\{X(t), t>0\}$ has the one-dimensional distribution\[\left\{F_{t}(x)=P(X(t) \leq x)=1-e^{-(x / t)^{2}}, x \geq 0, t>0\right\}\]Is this process weakly stationary?
The one-dimensional distribution of a stochastic process $\{X(t), t>0\}$ is\[F_{t}(x)=P(X(t) \leq x)=\frac{1}{\sqrt{2 \pi t} \sigma} \int_{-\infty}^{x} e^{-\frac{(u-\mu)^{2}}{2 \sigma^{2} t}} d
Let $X(t)=A \sin (\omega t+\Phi)$, where $A$ and $\Phi$ are independent, non-negative random variables with $\Phi$ uniformly distributed over $[0,2 \pi]$ and \(E(A)
Let $X(t)=A(t) \sin (\omega t+\Phi)$ where $A(t)$ and $\Phi$ are independent, non-negative random variables for all $t$, and let $\Phi$ be uniformly distributed over $[0,2 \pi]$.Verify:
Let $\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ be a sequence of real numbers, and $\left\{\Phi_{1}, \Phi_{2}, \ldots, \Phi_{n}\right\}$ be a sequence of independent random variables, uniformly
A modulated signal (pulse code modulation) $\{X(t), t \in(-\infty,+\infty)\}$ is given by\[X(t)=\Sigma_{-\infty}^{+\infty} A_{n} h(t-n)\]where the $A_{n}$ are independent and identically
Let $\{X(t), t \in(-\infty,+\infty)\}$ and $\{Y(t), t \in(-\infty,+\infty)\}$ be two independent, weakly stationary stochastic processes, whose trend functions are identically 0 and which have
Let $X(t)=\sin \Phi t$, where $\Phi$ is uniformly distributed over the interval $[0,2 \pi]$.Verify: (1) The discrete-time stochastic process $\{X(t) ; t=1,2, \ldots\}$ is weakly, but not
Let $\{X(t), t \in(-\infty,+\infty)\}$ and $\{Y(t), t \in(-\infty,+\infty)\}$ be two independent stochastic processes with trend and covariance functions\[m_{X}(t), m_{Y}(t) \text { and }
The following table shows the annual, inflation-adjusted profits of a bank in the years between 2005 to 2015 [in $\$ 10^{6}$ ].(1) Determine the smoothed values $\left\{y_{i}\right\}$ obtained by
The following table shows the production figures $x_{i}$ of cars of a company over a time period of 12 years (in $10^{3}$ ).(1) Draw a time series plot. Is the underlying trend function
Let $Y_{t}=0.8 Y_{t-1}+X_{t} ; t=0, \pm 1, \pm 2, \ldots$, where $\left\{X_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}$ is the purely random sequence with parameters $E\left(X_{t}\right)=0$ and
Let an autoregressive sequence of order $2\left\{Y_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}$ be given by\[Y_{t}-1.6 Y_{t-1}+0.68 Y_{t-2}=2 X_{t} ; \quad t=0, \pm 1, \pm 2, \ldots\]where
Let an autoregressive sequence of order $2\left\{Y_{t} ; t=0, \pm 1, \pm 2, \ldots\right\}$ be given by\[Y_{t}-0.8 Y_{t-1}-0.09 Y_{t-2}=X_{t} ; t=0, \pm 1, \pm 2, \ldots\]where \(\left\{X_{t} ;
Two dice are thrown. Their respective random outcomes are $X_{1}$ and $X_{2}$. Let $X=\max \left(X_{1}, X_{2}\right)$ and $Y$ be the number of even components of \(\left(X_{1}, X_{2}\right) .
Every day a car dealer sells $X$ cars of type 1 and $Y$ cars of type 2 . The following table shows the joint distribution $\left\{r_{i j}=P(X=i, Y=j) ; i, j=0,1,3\right\}$ of $(X, Y)$.(1)
Let $B$ be the upper half of the circle $x^{2}+y^{2}=1$. The random vector $(X, Y)$ is uniformly distributed over $B$.(1) Determine the joint density of $(X, Y)$.(2) Determine the marginal
Let the random vector $(X, Y)$ have a uniform distribution over a circle with radius $r=2$.Determine the distribution function of the point $(X, Y)$ from the center of this circle.
Tessa and Vanessa have agreed to meet at a café between 16 and 17 o'clock. The arrival times of Tessa and Vanessa are $X$ and $Y$, respectively. The random vector $(X, Y)$ is assumed to have a
Determine the mean length of a chord, which is randomly chosen in a circle with radius $r$. Consider separately the following ways how to randomly choose a chord:(1) For symmetry reasons, the
Matching bolts and nuts have the diameters $X$ and $Y$, respectively. The random vector $(X, Y)$ has a uniform distribution in a circle with radius $1 \mathrm{~mm}$ and midpoint (30mm ,
The random vector $(X, Y)$ is defined as follows: $X$ is uniformly distributed in the interval $[0,10]$. On condition $X=x$, the random variable $Y$ is uniformly distributed in the interval
Let\[f_{X, Y}(x, y)=c x^{2} y, 0 \leq x, y \leq 1\]be the joint probability density of the random vector $(X, Y)$.(1) Determine the constant $c$ and the marginal densities.(2) Are $X$ and $Y$

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