Questions and Answers of Introduction To The Mathematics

If we reverse the order of events in the definition of the limit of a sequence $\left(a_{n}\right)$, we get\[ \exists N \text { such that } \forall \varepsilon>0 \forall n \geq
A sequence $\left(a_{n}\right)$ is defined by\[ a_{1}=1 \text { and } a_{n+1}=\frac{a_{n}^{2}+2}{2 a_{n}} \forall n \geq 1 \](i) Prove that $\left(a_{n}\right)$ is a bounded sequence, and
Critic Ivor Smallbrain is sitting by the fire in his favourite pub, The Fox and Bounds. Also there are his friends Polly Gnomialle, Greta Picture, Gerry O'Laughing, Einstein, Hawking and celebrity
Let $\left(X_{t}\right)$ be a standard Brownian motion. Compute(a) $P\left[X_{5.4}>2\right]$;(b) \(P\left[X_{3.1}-X_{2.1}
Let $\left(X_{t}\right)$ be a Brownian motion with initial state 0 , drift rate 2 , and variance rate 4.Compute (a) $P\left[X_{8.2}-X_{4.5} \leq 6\right]$; (b) \(P\left[X_{1} \leq 3,
Write a program to simulate standard Brownian motion many times, and to return the proportion of the replications in which the process hits $M$ before $-N$, and compare your empirical results to
Extend the result of Example 2 for the probability $f(x)$ of hitting $M$ before $-N$ starting at $x$ to the case of Brownian motion with drift $\mu$ and variance rate 1.(Hint: Begin by
Write a simulator for non-standard Brownian motion with parameters $\mu$ and $\sigma$ analogous to the PlotSimulateBrownianMotion command in the section, which was designed for standard Brownian
Consider as in Example 4 a risky asset whose price behaves as $P_{1}(t)=p_{1} e^{X_{t}}$, where $\left(X_{t}\right)$ is a Brownian motion with drift rate $\mu$, variance rate $\sigma^{2}$,
Let $\left(X_{t}\right)$ be a standard Brownian motion with initial state 0 , and for a point $M>0$ let $T_{M}$ be the first time that the Brownian motion achieves a value of at least $M$.
Suppose that you have a parcel of land for sale, and you receive offers of $X_{t}$ at each time $t \in[0, T)$, where $\left(X_{t}\right)$ is a standard Brownian motion with initial state $x$.
Use formula (12) to solve explicitly for the optimal value of $w_{0}$ in terms of $a$, and show that the optimal portion of wealth invested in the non-risky asset increases as $a$ increases.
In the portfolio problem with the same choice of parameters $t=1, r=.05, \mu=.06, \sigma=.03$, how large should $a$ be so that it is optimal to keep at least half of the initial wealth in the
A geometric Brownian motion is a process $\left(Y_{t}\right)$ such that its $\log$ forms a standard Brownian motion. For such a process with initial state 1, find the probability density function
If $\left(X_{t}\right)$ is a standard Brownian motion, then the process $Y_{t}=M-\left|M-X_{t}\right|$ is called a Brownian motion with reflecting barrier at $M$. Explain the terminology, and
In advanced courses in stochastic processes, it is possible to define a stochastic integral with respect to a standard Brownian motion\[ \int_{a}^{b} X_{s} d W_{s} \]where $\left(X_{s}\right)$ is
Show that the probability that standard Brownian motion hits a point at a fixed distance $\epsilon>0$ from its starting point within a time interval of length $h$ is of the order $o(h)$. (Hint:
Let $f$ and $g$ be the following permutations in $S_{7}$ :\[ f=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 1 & 5 & 7 & 2 & 6 & 4 \end{array}\right), \quad
(a) List the numbers that occur as the orders of elements of $S_{4}$, and calculate how many elements there are in $S_{4}$ of each of these orders.(b) List all the possible cycle-shapes of even
Critic Ivor Smallbrain has been engaged for the prestigious role of dressing up as Father Christmas at Harrods this year. There, he will have to distribute $n+3$ toys to $n$ children. He must
For each of the following functions $f$, say whether $f$ is 1-1 and whether $f$ is onto:(i) $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=x^{2}+2 x$ for all \(x \in
The functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are defined as follows:\[ \begin{aligned} & f(x)=2 x \text { if } 0 \leq x \leq 1, \text { and } f(x)=1 \text { otherwise; } \\ & g(x)=x^{2}
Two functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are such that for all $x \in \mathbb{R}$,\[ g(x)=x^{2}+x+3, \quad \text { and }(g \circ f)(x)=x^{2}-3 x+5 \]Find the possibilities for
Let $X, Y, Z$ be sets, and let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be functions.(a) Given that $g \circ f$ is onto, can you deduce that $f$ is onto? Give a proof or a
(a) Find an onto function from $\mathbb{N}$ to $\mathbb{Z}$.(b) Find a 1-1 function from $\mathbb{Z}$ to $\mathbb{N}$.
(a) Let $S=\{1,2,3\}$ and $T=\{1,2,3,4,5\}$. How many functions are there from $S$ to $T$ ? How many of these are 1-1?(b) Let $|S|=m,|T|=n$ with $m \leq n$. Show that the number of 1-1
Prove Theorem 5(a).
Classify all states in the Markov chain with transition diagram above.
Prove Theorem 2.
Prove Lemma 1.
For the chain of Example 1, find paths of length 4 from every state to every other state in the set $\{4,5,6\}$, and check that 4 is the shortest path length with this property.
Verify condition (9).
Repeat Exercise 7 for the problem of Exercise 1 of Chapter 2, Section 3.
Prove Theorem 1.
If $F(x)=1-e^{-\lambda x}$ for $x>0$, and zero otherwise, find a formula for the $n$-fold convolution $F^{(n)}(t)$.
Let $G(x)$ be the c.d.f. of the continuous uniform distribution on $[0,2]$, and let $F(x)$ be the c.d.f. of the discrete uniform distribution on the set of states $\{1,2\}$. Find $G * F(t)$
Suppose that the inter-renewal times of a renewal process have the (discrete) Poisson distribution with parameter 2.Note that this means it is possible for successive renewal times to be the same.
The preparation of a report requires the efforts of three people, each one beginning after his predecessor finishes. When one report is finished, the next one is begun, etc. Suppose that person $i$
The elementary renewal theorem does not follow trivially from the renewal law of large numbers, because the convergence of a sequence of random variables $X_{1}, X_{2}, X_{3}, \ldots$ to a constant
A machine receives shocks at the times $T_{1}, T_{2}, T_{3}, \ldots$ of a renewal process, and incurs damage at a level of $D_{i}$ as a result of the $i^{\text {th }}$ shock. The damage from
Customers arrive to a single server according to a Poisson process with rate 3 per hour. Those customers who arrive when the server is busy are simply lost. The server requires a constant time of
Use Theorem 2 to find the renewal function in the case that the interrenewal distribution is deterministic with all of its weight on the point 1.5 .Exercises 9-12 lead the reader through a renewal
A delayed renewal process is similar to a renewal process, except that the c.d.f. $G$ of the first renewal time $T_{1}$ may be different from the common c.d.f. $F$ of the inter-renewal times
Given a Markov chain $\left(X_{n}\right)_{n \geq 0}$, argue that the times $S_{1}, S_{2}, S_{3}, \ldots$ of successive visits to a state $j$, starting from a state $i$, form a delayed renewal
There is a theorem from renewal theory called Blackwell's Renewal Theorem (see Ross ([52], Prop. 3.5.1) that is applicable to processes with integer-valued renewal times that are otherwise
Finally, with $\left(I_{n}\right)$ as in Exercise 11 and $N_{t}$ equal to the sum of the $I$ 's through $t$, appeal to the Renewal Law of Large Numbers to establish part (e) of Theorem 1 of
A machine begins in good running condition, lasts for an exponential length of time with rate $\lambda_{1}$, then breaks down. The repair of the machine lasts for an exponential period of time with
Three investments are available. The first pays fixed dividends of $\$ 100$ at the times of a renewal process whose inter-renewal distribution is exponential with parameter $1 / 5$. The second
A device is currently new. We replace it with an identical new device either when it breaks or at the fixed time $T$, whichever comes first. The lifetime of a device has the Weibull density:The
Consider a renewal process $\left(N_{t}\right)$ with discrete, geometric inter-renewal distribution:(a) Find the limit as $t \longrightarrow \infty$ of $N_{t} / t$.(b) Find explicitly the
A common stock currently sells at $\$ 20$ per share. The price changes by plus $\$ 1$ or minus $\$ 1$ (respectively with probabilities $p$ and $1-p$ ) at times \(T_{1}, T_{2}, T_{3},
(a) Compute the birth and death rates for the $M / M / 4 / 6$ queue.(b) Find the limiting probabilities, if $\lambda=4$ and $\mu=2$.(c) What effect does doubling the service rate have on the
Compute the traffic intensity of a single server queue in which arrivals form a Poisson process with rate 5, and service times have the Weibull distribution: $g(t)=2 t e^{-t^{2}}$, if $t>0$. Will
Find the limiting distribution of the $M / G / \infty$ queue.
Verify expressions (7) and (8) for the limiting mean queue length of the $M / M / 1 / N$ queue.
For an $M / M / 1 / N$ queue with arrival rate $\lambda=2$ and service rate $\mu=3$, find the smallest value of $N$ such that the limiting probability of two or fewer in the queue is less
In Example 2, set $\lambda=5.2$ and $\mu=1.3$. At least how many toll stations should there be so that $95 \%$ of the time in the long run, there are ten or fewer vehicles in queue?
Suppose that the barber of Example 3 is unlucky enough to have only one employee (himself), and no waiting space other than the chair used by the customer on which he is currently working. He wishes
A cattle rancher is preparing to brand his cattle. A single brander is working, who can finish one steer in exactly 5 seconds. If cattle arrive to the brander according to a Poisson process with rate
Write a Mathematica function that implements formula (14) for the repair time distribution of Example 4.Use it to plot the limiting distribution for states from 0 to 8 .Exercises 10-12 complete the
Let $\left(p_{n}\right)_{n=0,1,2, \ldots}$ form a discrete probability mass function on the non-negative integers. Define the probability generating function of this distribution as the function
Let $\Pi(z)$ be the probability generating function of the limiting distribution $\pi$ for the $M / G / 1$ queue, and let $Q(z)$ be the probability generating function of the distribution
Send $z \longrightarrow 1$ on both sides of the equation in Exercise 11, using L'Hopital's rule to evaluate the limit on the right side, to finish the proof that $\pi_{0}=1-ho$. (Hint: To
By viewing the $M / M / 1$ queue as a special case of the $G / M / 1$ queue, find the limiting distribution of the queue length, embedded at arrival instants. Compare this to the results of
Precisely one bit of information arrives to a processor every microsecond. The processor requires an exponentially distributed amount of time with rate $\ln (4)$ per microsecond to analyze a bit
Arrivals come to a single server queue with $\exp (2.5)$ service time distribution so that only two interarrival times, 0.8 and 1.2, are possible, occurring with equal likelihood. Find the limiting
Cars arrive to a state vehicle testing station according to a Poisson process with rate $\lambda$. It requires an exponential length of time with rate $\mu$ to test a car. But, cars arriving when
Consider an $M / M / 1 / N$ queue whose arrival and service rates are equal. Suppose that customers who join the queue receive a reward $R$ at the end of service, but pay at a rate of $C$ per
Let $\left(N_{t}\right)$ be a birth-death process with only two states, 0 and 1 . Denote by $\lambda$ the birth rate at state 0 , and $\mu$ the death rate at state 1 .(a) Write the Kolmogorov
Write a Mathematica function to take the output of the SimBirthDeathProcess command and find the proportion of time that the process was in each of the states it visited.
Give a plausibility argument for the forward equation for $P_{i 0}$ listed in (9).
In Example 2, for $\lambda=20$ and $\mu=5$, what number of beds $N$ is necessary so that the hospital can come within $\$ 5$ of their highest possible long-run expected profit per day?
A single individual in an essentially infinite population has a disease initially. Let $N_{t}$ be the cumulative number of individuals who have contracted the disease by time $t$; thus
Write the Kolmogorov forward equations for the process in Exercise 5, and verify that $P_{1 j}(t)$ listed in part (b) satisfies them.
In Example 3, for a family of size 10, what is the smallest value of the transmission constant $c$ such that the mean time until everyone has the cold is no more than 3 days?
A population begins with $n$ individuals, who die at the times of a birthdeath process with death rates $\mu_{i}, i=1, \ldots, n$. Write an expression for the mean and variance of the time until
A delicatessan has four service lines, each manned by one server. The food is not particularly good there, so customers who arrive when all servers are busy just decide to go to some other
The size of a fish population follows a birth-death process. Suppose that the birth rate when there are no fish is some positive constant $\lambda_{0}$ (i.e., migration to empty water is possible);
An electric generator can be running at one of three speeds at any time: high, low, or off. It cannot change directly from high to off, nor from off to high. When it is on low, the probability is \(2
In deriving the forward equations, we conditioned on the population at time $t$ in order to approximate $P_{i j}(t+h)$. Give a similar argument in which you condition on the population at time
For Exercise 4 of Section 4.1, the television inspection problem, find the probability, starting from each of the states $\mathrm{F}$ and $\mathrm{G}$, of being absorbed by each of the states
For the chain with transition matrix below, find the probabilities of absorption into each of the classes $\{1,2,3\}$ and $\{4,5\}$ starting from each of the transient states 6 and 7.Find also
Let $i, j$, and $k$ be states of a finite state Markov chain. If $i$ is transient and $j$ and $k$ are in the same recurrence class $C$, show that $f_{i j}=f_{i k}$.
Consider again Example 2, in which the firm gains or loses one million in assets at each instant. If we now let the gain probability $p$ be general, what is the smallest $p$ such that, starting
A retail clothing store has begun to issue credit cards in May. Of its card holders, 1000 have not paid the minimum payment in June. Company policy states that if an account has still not been paid
Find the limit as $n \longrightarrow \infty$ of $T^{n}$, where $T$ is the transition matrix of the Markov chain of Exercise 2 of Section 4.4. The transition matrix and diagram are shown below
A graduate school offers a 5-year Ph.D. program in mathematics. Its records show a $50 \%$ attrition rate between the first and second years, $40 \%$ between the second and third, $10 \%$
In system (2), define a matrix $F=\left(f_{i j}\right)$ to have a row for each transient state in the set $D$ of all transient states and a column for each recurrent state in a given recurrence
Some elementary texts on Markov chains present the following procedure for chains with absorbing states and transient states. Let $S$ be the submatrix of the full transition matrix $T$
Let $\left(X_{n}\right)$ be the Markov chain with transition matrix below. Find the expected number of visits from each transient state $i$ to each other transient state $j$. Find, for each
Calls arrive to a central telephone exchange according to $\mathrm{a}$ Poisson process with rate $\lambda=3.6$ per minute. Let $N_{t}$ be the total number of calls up to and including the
Verify the first line of (4).
A clock hangs precariously on a wall, falling occasionally. The clock ceases to work when it falls for the $k^{\text {th }}$ time. If falls occur according to a Poisson process with rate
If $\left(N_{t}\right)$ is a Poisson process with rate $\lambda$, find(a) $E\left[N_{t+s} \mid N_{t}\right]$(b) $E\left[N_{t} \cdot N_{t+s}\right]$
Suppose that patients arriving to a doctor's office form a Poisson process with rate $\lambda$ per hour. Given that there are $n$ patients during the 8 -hour day, what is the probability that
In order for a machine to continue functioning, each of two parts must work. One replacement is available for each of the parts. A part lasts for an exponentially distributed amount of time with
Suppose that cars traveling west on a two-lane highway pass a fixed point on the road at the times of a Poisson process with rate $\lambda_{1}$, and similarly the eastbound cars form a Poisson
In Example 3, suppose that $f$ is a non-negative function such that $\int_{0}^{\infty} f(t) d t$ is finite. Find the expected present value of all dividends earned throughout time.
A telephone customer service system can give an acceptable service level if it receives no more than two calls per minute. The supervisor of the system has a quality goal of giving acceptable service
Let $T_{1}$ and $T_{2}$ be the first two arrival times of a Poisson process $\left(N_{t}\right)$. Show that the joint conditional density of $T_{1}$ and $T_{2}$, given $N_{t}=2$, is (2/2

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