Questions and Answers of Probability And Stochastic Modeling

A small branch of a bank has the two tellers 1 and 2 . The service times at these tellers are independent and exponentially distributed with parameter $\lambda=0.4\left[\mathrm{~min}^{-1}\right]$.
Four weeks later Pumeza visits the same branch as in exercise 2.43 . Now the service times at tellers 1 and 2 are again independent, but exponentially distributed with respective parameters
An insurance company offers policies for fire insurance. Achmed holds a policy according to which he gets full refund for that part of the claim which exceeds $\$ 3000$. He gets nothing for a claim
Pedro runs a fruit shop. Mondays he opens his shop with a fresh supply of strawberries of $s$ pounds, which is supposed to satisfy the demand for three days. He knows that for this time span the
The probability density function of the random annual energy consumption $X$ of an enterprise [in $10^{8} \mathrm{kwh}$ ] is\[f(x)=30(x-2)^{2}\left[1-2(x-2)+(x-2)^{2}\right], \quad 2 \leq x \leq
The random variable $X$ is normally distributed with mean $\mu=5$ and standard deviation $\sigma=4$.Determine the respective values of $x$ which satisfy\[\begin{gathered} P(X \leq x)=0.5,
The response time of an average male car driver is normally distributed with mean value 0.5 and standard deviation 0.06 (in seconds).(1) What is the probability that his response time is greater than
The tensile strength of a certain brand of paper is modeled by a normal distribution with mean $24 p s i$ and variance $9[p s i]^{2}$.What is the probability that the tensile strength of a sample
The total monthly sick leave time of employees of a small company has a normal distribution with mean 100 hours and standard deviation 20 hours.(1) What is the probability that the total monthly sick
The random variable $X$ has a Weibull distribution with mean value 12 and variance 9 .(1) Calculate the parameters $\beta$ and $\theta$ of this distribution.(2) Determine the conditional
The random measurement error $X$ of a meter has a normal distribution with mean 0 and variance $\sigma^{2}$, i.e., $X=N\left(0, \sigma^{2}\right)$. It is known that the percentage of
If sand from gravel pit 1 is used, then molten glass for producing armored glass has a random impurity content $X$ which is $N(60,16)$-distributed. But if sand from gravel pit 2 is used, then
Let $X$ have a geometric distribution with\[P(X=i)=(1-p) p^{i} ; \quad i=0,1, \ldots ; 0
A random variable $X$ has distribution function\[F_{\alpha}(x)=e^{-\alpha / x} ; \alpha>0, x>0\](Frechét distribution).What distribution type arises when mixing this distribution with regard to
The random variable $X$ has distribution function\[F(x)=\frac{x}{x+1}, x \geq 0\]Check whether there is a subinterval of $[0, \infty)$ on which $F(x)$ is $D F R$ or $I F R$.
Check the aging behavior of systems whose lifetime distributions have(1) a Frechét distribution with distribution function $F(x)=e^{-(1 / x)^{2}}, x>0$ (sketch its failure rate), and(2) a power
Let $F(x)$ be the distribution function of a nonnegative random variable $X$ with finite mean value $\mu$.(1) Show that the function $F_{s}(x)$ defined by\[F_{S}(x)=\frac{1}{\mu}
Let $X$ be a random variable with range $\{1,2, \ldots\}$ and probability distribution\[P(X=i)=\left(1-\frac{1}{n^{2}}\right) \frac{1}{n^{2(i-1)}} ; i=1,2, \ldots\]Determine the $z$-transform
Determine the Laplace transform $\hat{f}(s)$ of the density of the Laplace distribution with parameters $\lambda$ and $\mu$:\[f(x)=\frac{1}{2} \lambda e^{-\lambda|x-\mu|}, \quad-\infty
Table 8.1 No . 2 records the solution to the initial value problem (8.6). Explain how this result follows from our considerations regarding (8.3) and (8.7).Data From Equation (8.3)Data From Problem
Let $\left(X_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}$ be a real-valued process whose paths $t \mapsto X_{t}(\omega)$ are (a.s.) right continuous and have finite left-hand limits (this
Let $\tau:=\tau_{(-a, b)^{c}}^{\circ}$ be the first entrance time of a $\mathrm{BM}^{1}$ into the set $(-a, b)^{c}$.a) Show that $\tau$ has finite moments $\mathbb{E} \tau^{n}$ of any order
Let $\left(X_{t}\right)_{t \geqslant 0}$ be a $d$-dimensional Feller process and let $f, g \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)$. Show that the function \(x \mapsto
Let $\left(P_{t}\right)_{t \geqslant 0}$ be the transition semigroup of a $\mathrm{BM}^{d}$ and denote by \(L^{p}, 1 \leqslant p
Let $\left(T_{t}\right)_{t \geqslant 0}$ be a Markov semigroup given by $T_{t} u(x)=\int_{\mathbb{R}^{d}} u(y) p_{t}(x, d y)$ where $p_{t}(x, C)$ is a kernel in the sense of Remark 7.6. Show
Show that $p_{t_{1}, \ldots, t_{n}}^{x}\left(C_{1} \times \cdots \times C_{n}\right)$ of Remark 7.6 define measures on $\mathscr{B}\left(\mathbb{R}^{d \cdot n}\right)$.Data From 7.6 Remark 7.6
Let $\left(T_{t}\right)_{t \geqslant 0}$ be a Feller semigroup, i.e. a strongly continuous, positivity preserving sub-Markovian semigroup on
Complete the following alternative argument for Example 7.16. Assume that $\left(u_{n}\right)_{n \geqslant 1} \subset \mathcal{C}_{\infty}^{2}(\mathbb{R})$ such that \(\left(\frac{1}{2}
Let $(A, \mathfrak{D}(A))$ be the generator of a $\mathrm{BM}^{d}$. Adapt the arguments of Example 7.25 and show that \(\mathcal{C}_{\infty}^{2}\left(\mathbb{R}^{d}\right) \varsubsetneqq
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and consider the two-dimensional process $X_{t}:=\left(t, B_{t}\right), t \geqslant 0$.a) Show that \(\left(X_{t}\right)_{t
A Poisson process is a real-valued stochastic process $\left(N_{t}\right)_{t \geqslant 0}$ such that $N_{0}=0$, $N_{t}-N_{s} \sim N_{t-s}$ and for $t_{0}=00$. In particular,
Let $C$ denote Cantor's discontinuum which is obtained if we remove recursively the open middle third of any remaining interval:\[[0,1] \leadsto\left[0, \frac{1}{3}\right] \cup\left[\frac{2}{3},
Denote by $\mathcal{H}^{1}$ the Cameron-Martin space (13.2) and by $\mathcal{H}_{\circ}^{1}$ the set defined in (13.3).a) Show that $\mathcal{H}^{1}$ is a Hilbert space with the canonical norm
a) Show that b) Show that Let B, H, T be real random variables on the same probability space. We write "", and "H", for "independent", and "conditionally independent given H", respectively.
Show that the limit (15.19) does not depend on the approximating sequence.Data From (15.19) L(P)-lim fn Bt 11-00 (15.19)
Use Theorem 15.15.c) to show that the stochastic integrals for the right and left continuous simple processes \(f(t, \omega):=\sum_{j=1}^{n} \phi_{j-1}(\omega) \mathbb{1}_{\left[s_{j-1},
Let $\left(B_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}$ be a one-dimensional Brownian motioin and $\tau$ a stopping time. Show that \(f(s, \omega):=\mathbb{1}_{[0, T \wedge \tau(\omega))}(s), 0
Show that the process $f^{2} \bullet\langle Mangle_{t}:=\int_{0}^{t}|f(s)|^{2} d\langle Mangle_{s}$ appearing in Theorem 17.9.b) is adapted.Data From 17.9 Theorem 17.9 Theorem. Let (Mt, ter be in
The following exercise contains an alternative proof of Itô's formula (18.1) for a one-dimensional Brownian motion $\left(B_{t}\right)_{t \geqslant 0}$.a) Let \(f \in
a) Use the (two-dimensional, deterministic) chain rule $d(F \circ G)=F^{\prime} \circ G d G$ to deduce the formula for integration by parts for Stieltjes integrals:\[\int_{0}^{t} f(s) d g(s)=f(t)
Show that $\beta_{t}=\int_{0}^{t} \operatorname{sgn}\left(B_{s}\right) d B_{s}$ is a $\mathrm{BM}^{1}$.Use Lévy's characterization of a $\mathrm{BM}^{1}$, Theorem 9.13 or 19.5.Data From
State and prove a $d$-dimensional version of the Burkholder-Davis-Gundy inequalities (19.21) if $p \in[2, \infty)$.Use the fact that all norms in $\mathbb{R}^{d}$ are equivalentData From
Let and assume that Show that f.g L(R.)
Let $f \in L^{2}\left(\mathbb{R}_{m}^{+}\right), g \in L^{2}\left(\mathbb{R}_{n}^{+}\right)$and assume that the functions are symmetric: $f=\widehat{f}$, $g=\widehat{g}$. We define the
Let $X=\left(X_{t}\right)_{t \geqslant 0}$ be the process from Example 21.2. Show that $X$ is a Gaussian process with independent increments and find \(C(s, t)=\mathbb{E} X_{s} X_{t}, s, t
Let $B_{t}=\left(b_{t}, \beta_{t}\right)$ be a $\mathrm{BM}^{2}$. Solve the $\mathrm{SDE}$\[X_{t}=x+b \int_{0}^{t} X_{s} d s+\sigma_{1} \int_{0}^{t} X_{s} d b_{s}+\sigma_{2} \int_{0}^{t} X_{s}
Show that in Example 21.7\[X_{t}^{\circ}=\exp \left(-\int_{0}^{t}\left(\beta(s)-\delta^{2}(s) / 2\right) d s\right) \exp \left(-\int_{0}^{t} \delta(s) d B_{s}\right)\]and verify the expression given
Show that Remark 22.5.a) remains valid if we replace $f(x)$ by $f(s, x)$. 22.5 Remark. a) The Stratonovich integral satisfies the classical chain rule. Indeed, if fe e(R) and dX = (t) dB + b(t)
Show that Example 22.7 remains valid if we assume that $\sigma(0)=0$ and $\sigma(x)>0$ for $x eq 0$. 22.7 Example. Let (Bt)to be a BM and assume that = c(R) is either strictly positive or
Let $B=\left(B_{t}\right)_{t \in[0,1]}$ be a $\mathrm{BM}^{1}$ and set $W=\left(W_{t}\right)_{t \in[0,1]}$ where $W_{t}:=B_{t}-t B_{1}$.a) Show that $W$ is a mean zero Gaussian process and
Let $\left(\mathcal{C}_{(\mathrm{o})}, \mathscr{B}\left(\mathcal{C}_{(\mathrm{o})}\right), \mu\right)$ be the canonical Wiener space (we assume $d=1$ ). We will now consider the space
We continue with the set-up introduced in Problem 7.Denote by $\left(\mathcal{C}_{(0)}, \mathscr{B}\left(\mathcal{C}_{(0)}\right), \mu\right)$ the canonical Wiener space and set
Let $B=\left(B_{t}\right)_{t \geqslant 0}$ be a canonical $\mathrm{BM}^{1}$ on Wiener space \((\Omega, \mathscr{A}, \mathbb{P})=\left(\mathcal{C}_{(0)}, \mathscr{B}\left(\mathcal{C}_{(0)}\right),
What is Stats?
Let $\left(\mathscr{F}_{t}\right)_{t \geqslant 0}$ be an admissible filtration for the Brownian motion $\left(B_{t}\right)_{t \geqslant 0}$. Mimic the proof of Lemma 2.14 and show that for
Assume that the processes in Lemma 5.7 or in Lemma 5.8 are only "almost surely right continuous", resp. "almost surely continuous". Identify all steps in the proofs where this becomes relevant and
Let $\left(B_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ such that $\mathscr{F}_{0}$ contains all measurable null sets and let $\tau$ be a stopping time. Show that
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ with transition semigroup $\left(P_{t}\right)_{t \geqslant 0}$. Show, using arguments similar to those in the proof of
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{d}$ and set $u(t, x):=P_{t} u(x)$. Adapt the proof of Proposition 7.3.g) and show that for \(u \in
a) Let $d(x, A):=\inf _{a \in A}|x-a|$ be the distance between the point $x \in \mathbb{R}^{d}$ and the set $A \in \mathscr{B}\left(\mathbb{R}^{d}\right)$. Show that $x \mapsto d(x, A)$ is
Show that Dynkin's formula (7.30) in Proposition 7.31 follows from Theorem 7.30.Data From Theorem 7.30 Ex. 7.18 7.30 Theorem. Let (Xt, F)to be a Feller process on Rd with transition semigroup
Assume that $L: \mathcal{C}_{c}^{\infty}\left(\mathbb{R}^{d}\right) \rightarrow \mathcal{C}\left(\mathbb{R}^{d}\right)$ is a local operator (cf. Definition 7.36) which is almost positive, i.e. for
Let $f$ be the initial value in the problem considered in Lemma 8.1 and write $u(t, x):=$ $P_{t} f(x)=\mathbb{E} f\left(B_{t}+x\right)$ where $\left(B_{t}\right)_{t \geqslant 0}$ is a BM \({
Assume that in Lemma 8.1 the initial datum satisfies $f \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)$ but not necessarily $f \in \mathfrak{D}(A)$. Consider (8.3) with $P_{\epsilon} f$
Complete the approximation argument for Lévy's arc-sine law from $\S 8.9$ :a) Show, by a direct calculation, that $v_{n, \lambda}(x)$ converges as $n \rightarrow \infty$. Conclude from (8.16)
Show Theorem 8.5 with semigroup methods.Observe that $A \int_{0}^{t} P_{s} g d s=P_{t} g-g=\frac{d}{d t} \int_{0}^{t} P_{s} g d s-g$.Data From Theorem 8.5 8.5 Theorem. Let (Bt)to be a BMd and g =
Find the solution to the Dirichlet problem in dimension $d=1: u^{\prime \prime}(x)=0$ for all $x \in(0,1), u(0)=a, u(1)=b$ and $u$ is continuous in [0,1]. Compare your findings with Wald's
Use Lemma 7.33 and give an alternative derivation of the result of Lemma 8.10.Data From Lemma 7.33Data From Leema 8.10 7.33 Lemma. Let (X,,F)o be a Feller process with a right continuous filtration,
Show that Lemma 8.10 remains true for any $d$-dimensional Feller process $X_{t}$ with continuous paths and generator \(L=\sum_{j, k=1}^{d} a_{j k}(x) \partial_{j} \partial_{k}+\sum_{j=1}^{d}
Let $g:[0, \infty) \times \mathbb{R}^{d} \rightarrow \mathbb{R}$ be a bounded continuous function such that $g(t, \cdot)$ is $\kappa$-Hölder continuous with a Hölder constant which does not
Use the LIL (Corollary 12.2) to give an alternative proof for the fact that a onedimensional Brownian motion oscillates in every time interval $[0, \epsilon]$ infinitely often around its starting
Let $d \geqslant 2$. A flat cone in $\mathbb{R}^{d}$ is a cone in $\mathbb{R}^{d-1}$. Adapt the argument of Example 8.18.e) to show the following useful regularity criterion for a
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a one-dimensional Brownian motion. Show that \[\operatorname{dim} B^{-1}(A) \leqslant \frac{1}{2}+\frac{1}{2} \operatorname{dim} A \quad \text { a.s. for
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and denote by $g_{t}$ the last zero before time $t>0$.a) Use Theorem 11.25 to give a further proof of Lévy's arc-sine law
Let $\left(B_{t}\right)_{t \geqslant 0}, d_{t}$ and $g_{t}$ be as in Corollary 11.26. Define\[L_{t}^{-}:=t-g_{t} \quad \text { and } \quad L_{t}:=d_{t}-g_{t}\]Find the laws of \(\left(L_{t}^{-},
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Apply Doob's maximal inequality (A.13) to the exponential martingale \(M_{t}^{\xi}:=\exp \left(\xi B_{t}-\frac{1}{2} \xi^{2}
Let $X=\left(X_{t}\right)_{t \geqslant 0}$ be a one-dimensional process satisfying (B0), (B1), (B2). Assume, in addition, that $X$ is continuous in probability, i.e. \(\lim _{t \rightarrow 0}
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $\tau$ some a.s. finite stopping time such that $\left|B_{t \wedge \tau}\right|, t \geqslant 0$, is bounded. Show that
Let $f \in \mathcal{S}_{T}$ be a simple process and $M \in \mathcal{M}_{T}^{2, c}$. Show that the definition of the stochastic integral $\int_{0}^{T} f(s) d M_{S}$ (cf. Definition 15.9) does
Use the fact that any continuous, square-integrable martingale with bounded variation paths is constant (cf. Proposition 17.2) to show the following: \(\langle f \cdot
The quadratic covariation of two continuous $L^{2}$ martingales $M, N \in \mathcal{M}_{T}^{2, c}$ is defined as in the discrete case by the polarization formula (15.8).Let \(\left(B_{t}\right)_{t
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and \(f \in \operatorname{BV}[0, T], TData From 13.5 Paragraph G (w) = $(s) dw(s) = (1)w(1) - [w(s) d(s), w C(o)[0, 1], 1)-\ - is
Adapt the proof of Proposition 15.18 and prove that we also have \[\lim _{|\Pi| \rightarrow 0} \mathbb{E}\left[\sup _{t \leqslant T}\left|\int_{0}^{t} f(s) d B_{s}-\sum_{j=1}^{n}
Let $\tau$ be a stopping for a $\mathrm{BM}^{1}\left(B_{t}\right)_{t \geqslant 0}$ and define the stochastic intervalthatApproximate $\tau$ by stopping times with finitely many values as in
Show that Proposition 17.2 and Corollary 17.3 remain valid for local martingales with continuous paths (cf. Definition 16.7).Data From Proposition 17.2Data From Corollary 17.3 17.2 Proposition. Let
Show that the limits (18.5) and (18.6) actually hold in $L^{2}(\mathbb{P})$ and uniformly for $T$ from compact sets.To show that the limits hold uniformly, use Doob's maximal inequality. For the
Let $\left(N_{t}\right)_{t \geqslant 0}$ be a Poisson process with intensity $\lambda=1$ (see Problem 10.1 for the definition). Show that for \(\mathscr{F}_{t}^{N}:=\sigma\left(N_{r}: r \leqslant
Let XtXt be as in Problem 19.5 and set a) Use Problem 19.5 to find the probability density of ˆτbτ^b if α,b>0α,b>0.b) Find and , respectively.Data From Problem 19.5 b := inf{t 0 : X
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$, denote by $\mathcal{H}_{T}^{2}$ the space used in Lemma 19.10 and pick $\xi_{1}, \ldots, \xi_{n} \in \mathbb{R}$ and
Show that, in Theorem 19.17, the quadratic variation $\langle Mangle_{t}$ is a $\mathscr{G}_{t}$ stopping time.Direct calculation, use Lemma 19.16.c) and A. 15Data From Theorem 19.17 19.17
We have seen in Lemma 19.27.a) that $\operatorname{supp}\left[d L_{t}^{0}(\omega)\right] \subset\left\{t \geqslant 0: B_{t}(\omega)=0\right\}$ for almost all $\omega$. Show that
The proof of Theorem 19.29 uses, implicitly, the following beautiful result due to Skorokhod [239] which is to be proved:Lemma. Let $b:[0, \infty) \rightarrow \mathbb{R}$ be a continuous function
Show that in Lemma 19.32.a) the following stronger assertion holds: $\left(L_{t}\right)_{t \geqslant 0}$ is an additive functional, i.e. $L_{t+s}=L_{t}+L_{s} \circ \theta_{t}$ holds for all \(s,
Let $\left(X_{t}, \mathscr{G}_{t}\right)$ be an adapted, real-valued process with right continuous paths and finite left limits. Assume that \(\mathbb{P}\left(X_{t}-X_{s} \in A \mid
Show that the definition of the double Itô integral for off-diagonal simple functions (Definition 20.4) is independent of the representation of the simple function.Data From Definition 20.4 20.4
Repeat the calculation from the end of Example 20.5 for a general measure $\mu$. What happens on the diagonal?Data From Example 20.5 20.5 Example. Let us return to Example 20.1. If II = {0 = to
Show that the definition of the iterated Itô integral for $f \in L^{2}\left(\mathbb{R}_{+}^{2}\right)$ (Definition 20.9) is independent of the approximating sequence.Data From 20.9 Definition
Prove (20.20), i.e. show that $\mathbb{E}\left[B_{T}^{2 n} e^{-B_{T}^{2}}\right]=\frac{(2 n-1) ! !}{\sqrt{2 T+1}}\left(\frac{T}{2 T+1}\right)^{n}$. Here, $(-1) ! !:=1$, \((2 n-1) ! !=1 \cdot 3
Verify the claim made in Example 21.9 using Itô’s formula.Derive from the proof of Lemma 21.8 explicitly the form of the transformation and the coefficients in Example 21.9.Integrate the condition
Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Use Lemma 21.10 to find the solution of the following SDE:\[d X_{t}=\left(\sqrt{1+X_{t}^{2}}+\frac{1}{2} X_{t}\right) d

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