Questions and Answers of Probability And Stochastic Modeling

A \(\mathbb{R}\)-valued stochastic process \(\left(X_{t}ight)_{t \geqslant 0}\) such that \(\mathbb{E}\left(X_{t}^{2}ight)
Let \(\left(B_{t}ight)_{t \in[0,1]}\) and \(\left(\beta_{t}ight)_{t \in[0,1]}\) be independent one-dimensional Brownian motions. Show that the following process is again a Brownian motion:\[W_{t}:=
Fractional Brownian motion (fBM, \(\alpha\)-fBM) \(\left(B_{t}^{\alpha}ight)_{t \in \mathbb{R}}\) with Hurst index \(H=\alpha / 2 \in(0,1]\) is a Gaussian process with the following mean and
Use the Lévy-Ciesielski representation \(B(t)=\sum_{n=0}^{\infty} G_{n} S_{n}(t), t \in[0,1]\), to obtain a series representation for \(X:=\int_{0}^{1} B(t) d t\) and find the distribution of \(X\).
Let \(\left(G_{n}ight)_{n \geqslant 0}\) be a sequence of iid Gaussian \(\mathrm{N}(0,1)\) random variables, \(\left(\phi_{n}ight)_{n \geqslant 0}\) be a complete ONS in \(L^{2}([0,1], d t), B \in
Let \(\phi_{n}=H_{n}, n=0,1,2, \ldots\), be the Haar functions. The following steps show that the full sequence \(W_{N}(t, \omega):=\sum_{n=0}^{N-1} G_{n}(\omega) S_{n}(t), N=1,2, \ldots\), converges
Let \((S, d)\) be a complete metric space equipped with the \(\sigma\)-algebra \(\mathscr{B}(S)\) of its Borel sets. Assume that \(\left(X_{n}ight)_{n \geqslant 1}\) is a sequence of \(S\)-valued
Let \(\left(X_{t}ight)_{t \geqslant 0}\) and \(\left(Y_{t}ight)_{t \geqslant 0}\) be two stochastic processes which are modifications of each other. Show that they have the same finite dimensional
Let \(\left(X_{t}ight)_{t \in I}\) and \(\left(Y_{t}ight)_{t \in I}\) be two processes with the same index set \(I \subset[0, \infty)\) and state space. Show that\[X, Y \text { indistinguishable }
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a real-valued stochastic process with exclusively continuous sample paths. Assume that \(\left(B_{q}ight)_{q \in \mathbb{Q} \cap[0, \infty)}\) satisfies
Let \(X^{\prime}, X^{\prime \prime}, Z\) be random vectors such that . Show that \(X^{\prime}, X^{\prime \prime}, Z\) are independent. X'X" and (X', X") ILZ.
Show that there is a stochastic process \(\left(W_{t}ight)_{t \in \mathbb{R}}\) satisfying (B0)-(B4). Glue together, back-to-back, two independent Browninan motions .
Using a probability theory argument, show that the sum of all elements of any covariance matrix is non-negative.
This exercise concerns the Pareto distribution that proved to be a good model for many real variables such as the sizes of towns, files of Internet traffic, meteorites, sand particles, etc.Consider a
Suppose that for r.v.’s X1 and X2 with respective d.f.’s F1(x) and F2(x), it is true that F1(x) ≤ F2(x) for all x’s. Such a relation is referred to as the first stochastic dominance (FSD),
Compute the mean of the standard exponential distribution using formula (1.3.2) and compare your calculations with what we did in Section 2.2. = (1-F(x))dx. E{X} = (1.3.2)
The claim department of a company is receiving claims. The time between the arrivals of two consecutive claims (an interarrival time) is an exponential r.v., and the probability that this time is
(a) Give a common-sense explanation why values of S2 are not equally likely.(b) Carry out calculations in (4.1.4). = = fs (x - y)dy, fs3 (x) = (4.1.4)
(a) Give an alternative proof (without m.g.f’s) of the result of Example 2.3-2 using Example 6.4.2-2.(b) Generalize the result of Example 2.3-2 to the case where the Xi’s are still exponential
Generalizing Exercise 5-c for r.v.’s Xi having a mean of m and a variance of σ2, find the limitComment on your result and connect it with the LLN.Exercise 5-cLet X1,X2, ... be a sequence of
Using software, say Excel, provide a worksheet demonstrating the accuracy of the normal approximation of the Poisson distribution for large λ.
prove that the normalized r.v.is asymptotically normal as ν→∞. Zav = (Zav - X) / () a a
Consider a sequence of independent standard exponential r.v.’s X1,X2, ..., and the r.v. Sn = X1+...+Xn. Let Γaν(x) be the d.f. of a Γ-r.v. with parameters (a, ν), and as usual, faν(x) be the
Is it true that if Corr(X1,X2) = 1, then X1 = cX2 for a c > 0? Answer the same question under the additional condition E{X1} = E{X2} = 0. Let mi = E{Xi}, i = 1,2. Is it true that if Corr(X1,X2) = 1,
Let a r.v. X have the density(a) Find c, and the probabilities P(0.5 ≤ X ≤ 1), P(X = 1), P(X > 3), and P(X (b) Find and graph the distribution function. (c) Find E{X} and E{X7}. (d) Find
Let a r.v. X have the densityFind f(x) for all other x’s.
Let a random variable X have the (cumulative) distribution function(a) Find F(x) for x 1.(b) Find P(0.5 ≤ X ≤ 1), P(X = 1), P(X = 0.25).(c) Find the density.
Let X be the r.v. from Exercise 1. Find the d.f. and density of the r.v. Y = X3.Exercise 1Let a r.v. X have the density
Let X be uniform on [-1,1]. Find the d.f. and density of the r.v. Y = arcsinX. Do the same for the r.v. Z = |Y|.
Let the d.f. F(x) = 1−cosx for 0 ≤ x ≤ π/2. Check that this function has all properties of d.f.’s. What values would a corresponding r.v. assume? Figure out what F(x) equals for x /∈ [0,
Let a r.v. X have the density(a) Explain why P(X /∈ [0,2])= 0. Without any calculations, find P(0 ≤ X ≤ 1) and P(1 ≤ X ≤ 2).(b) Reasoning heuristically, explain why E{X} should be equal
A random variable X has the density f (x)=0 for x for x ∈ [k,k + 1) and k = 0,1, ... . Graph f (x). What is P(k ≤ X ≤ k+1) and P(k ≤ X ≤ k+ 1/2|k ≤ X ≤ k+1)?Show that the distribution
Consider the inventory problem.(a) Explain why the solution to (1.4.3) is non-decreasing when δ is decreasing. Interpret this fact in the context of the problem.(b) Find the optimal size s of the
Graph the density and the distribution function, and find the expectation and standard deviation for the distribution uniform on [−3,2]. Calculate in mind the probability “to get into” the
Graph the density and the distribution function of the exponential distribution.
The continuous distribution with a density f(x) is said to be symmetric if for a number s called a center of symmetry, and for any x > 0,(a) Which distributions considered in Sections 1.1 and 2
A number μ is called a median for a continuous r.v. X if P(X μ) = 1/2.(a) Show that for any symmetric distribution, its median, center of symmetry, and mean (if it exists) coincide. (Advice:
A number q is called a mode of a continuous distribution with a density f (x) if f (x) attains its maximum at q. (So, q may be called the most probable value.)(a) Does a mode have to be unique? If
Generalizing (1.3.2), prove that for any continuous r.v. with a d.f. F(x) and a finite expected value,What would we have for a non-positive r.v.? In Chapter 7, we show that (6.2) is true for any
Let X be an exponential r.v., and P(X > 1) = 0.2. Not computing the parameter a, find P(X > 3) and P(X > 1.5).
Ann is the first in line for a service. There are three service counters (which are busy), the service times are independent and have the same exponential distribution. Find the probabilities that
Let a r.v. X be exponentially distributed, and E{X}= 2. Write the density. Find the variance. Find P(X ≤ 4| X > 1), P(3 ≤ X ≤ 4| X > 1).
For a random variable X having the-lack-of-memory property, let P(X > 2) = 3/4. Find the parameter a, E{X}, and the standard deviation of X.
Suppose you have found a median and mode of the Γ-distribution with the scale parameter a = 1. How to find the same characteristics for an arbitrary a > 0? Justify the answer.
For any distribution with mean m = 0 and standard deviation σ, the ratio σ/m is called a coefficient of variation (c.v.).(a) Will the c.v. change if we multiply the r.v. by a number c = 0?(b) Does
Let Xa,ν denote a r.v. having the Γ-distribution with parameters a and ν.(a) What is the difference between P(X1,ν < x) and P(aXa,ν < x)?(b) For which ν does the Γ-density f1,ν(x) converges
When solving problems below, it makes sense to keep in mind (2.4.3).(a) Let X be standard normal. What is the distribution of −X? Generalize the problem considering a normal r.v. with zero mean.
Let a r.v. Y be normal with an expectation of 3, and a variance of 16. Write a formula for P(−2 ≤ Y ≤ 6). Estimate this probability using Table 3 from the Appendix.
Let X be a normal r.v. with mean 12 and variance 4. Write the normalized r.v. Estimate c such that P(X < c) = 0.2.
Let X1 and X2 be continuous r.v.’s with the joint density f (x1,x2) = 1/2 if |x1|+|x2| ≤ 1, and = 0 otherwise. Explain why f(x1,x2) is a density. Find P(X21 +X22 ≤ 1/2).
Let a three-dimensional r.vec. X be uniformly distributed in a unit ball with a center at the origin. Find the d.f., expectation and variance of |X|, the length of X.
Let the two-dimensional density of a r.vec.wherethe length of the vector x. Find P(|X| > 2), E{X1}.
Figure out whether the coordinates X1,X2 are independent in(a) Exercise 32,Let the two-dimensional density of a r.vec.wherethe length of the vector x. Find P(|X| > 2), E{X1}.(b) Example 3-1.Let
A random vector (X,Y) has the joint density functionShow that f (x,y) is a density. Guess whether X and Y are independent. Justify your answer. Write the marginal densities.
Suppose that a r.vec. X = (X1,X2) has the distribution uniform on the circle (not a disk!) x21 +x21 = r2 for a fixed r. This means that the vector X takes values (x1,x2) only from the circle
A random vector (X,Y) has the joint density functionFind c. Guess whether X and Y are independent. Justify your answer. Write the marginal densities.
Let us revisit the record value problem of Section 3.2.3. Make sure that all reasonings there remain true in the case where X’s are continuous. Show that in this caseand that P(N for n = 1,2, ...
(a) Can we switch f1 and f2 in (4.1.1)?(b) How should we change formula (4.1.1) to get the density of the r.v. X1−X2?
Find and graph the probability density of the sum of independent r.v.’s X1 and X2 if(a) X1 and X2 are exponential with E{X1} = 1 and E{X2} = 1/2, respectively;(b) X1 and X2 are uniform on
Let X1,X2 be independent standard exponential r.v.’s. Show that the density of the r.v. X1−X2 isGraph it. The distribution with this density is called two-sided exponential. Show also that the
A signal, when going through an electronic device, is distorted, which amounts to adding a random distortion error X. Suppose that X is normal with zero mean and a standard deviation of ε. The
Let X and Y be independent continuous random variables with the densities eπ(x−1)2 andrespectively. Using Table 3 from the Appendix, estimate P(2 ≤ S ≤ 4), where S = X +Y.
Using the convolution formula, show that the sum of two independent standard normal r.v.’s is normal with mean 0 and variance 2 (which is consistent with Proposition 5). (Advice: First, show
Let Zn be a Γ-r.v. with a scale parameter of a and parameter ν equal to an integer n. Show that Zn may be represented as Y1+...+Yn, where Y’s are independent exponential r.v.’s with parameter a.
Let r.v.’s X1 and X2 be both Γ-distributed with a common scale parameter a and “essential” parameters ν1 and ν2; ν1 > ν2. Show that X1 dominates X2 in the sense of the FSD. Can we claim
Consider.(a) What is the mean of the total waiting time E{S4} ? (b) What is the probability that S4 will be equal to E{S4} ?(c) When computing P(S4 4}), can we set a = 1? Using software, compute
Let Sn = X1 +...+Xn, where the X’s are independent, and Xi has the Γ-distribution with parameters (1, (1/2)i). To which distribution is the distribution of Sn close for large n? Write limn→∞
The densities of independent r.v.’s X1 and X2 are f1(x) = C1x5e−4x and f2(x) = C2x8e−4x, respectively, where C1 and C2 are constants.(a) Do we need to calculate these constants in order to find
Using formula (5.1.5), prove that, if independent r.v.’s X1 and X2 are standard normal and S = X1 + X2, then the conditional distribution of X1 given S = s, is normal with a mean of s/2 and a
Let X1 and X2 have the Γ-distributions with the common scale parameter a = 1 and essential parameters ν1 and ν2, respectively. Let S = X1+X2. Given S = s, the r.v. X1 takes on values from [0,
Let the joint density of (X,Y) be f (x,y) = x+y for 0 < x ≤ 1,0 ≤ y ≤ 1, and =0 otherwise. Find the conditional density f (y|x), and E{Y |X}. At which x does the regression function m(x) = E{Y
Let the joint density of (X,Y) beotherwise. Find the conditional density f(y|x) and E{Y|X}.
A phone representative of a company is waiting for the next customer’s call. Suppose that men and women call independently, the waiting time for the next man’s call is exponentially distributed
Suppose that Rich is the third in line. Given that Rich has been waiting t minutes, find the distribution of the waiting time for being the first in line. Find the conditional mean. (Advice: The
In Example 5.2-4, set m1 = m, m2 = 1 and graph P(ξ2 > ξ1) as a function of m. (Advice: Pay attention to the behavior in a neighborhood of m = 1.)Example 5.2-4
R.v’s ξ1, ξ2 are independent and exponential with a parameter a. Find the distribution (say, the density) of the r.v. Z = ξ1/ξ2. (Advice: First, realize that the distribution of Z should not
Let ξ1, ..., ξn be i.i.d. r.v.’s with a continuous distribution. Explain why the r.v.is well defined; that is, we should not worry about the case where the denominator is zero. Find E{X1} and
Let X1 and X2 be i.i.d. r.v.’s. Assume that we know the values of Xmin = min{X1,X2}, and Xmax = max{X1,X2}. What can we say about the expected value of, for instance, X1 given this information?
Let Θ be a r.v. whose density f(θ) = 3θ2 for 0 ≤ θ ≤ 1, and f(θ) =0 otherwise. Let N be the number of the first success in a sequence of independent trials with the probability of success
Suppose that we know that the numbers a and b are chosen independently and at random from the interval [0,1]. Then it makes sense to choose X taking on values from [0,1]. Find P(A) for X uniform on
Consider a r.v. X with the d.f. F(x) graphed in Fig. 15.(a) Find P(1 ≤ X ≤ 4), P(1 (b) Compute in mind P(X (c) Specify the decomposition (1.3.3). Find E{X}.
Find a 0.2-quantile of a r.v. X taking values 0,3,7 with probabilities 0.1,0.3, 0.6, respectively.
Find y-quantiles of a r.v. X that(a) Uniform on [a,b];(b) Exponential with a mean of m.
Let qy be a y-quantile of the (m, σ2)-normal distribution. Show that qy = m + qysσ, where qys is a y-quantile for the standard normal distribution. When qy m? Consider particular cases y = 0.5,
Denote by qy(X) a y-quantile of a r.v. X. Let X have a continuous d.f. F(x) strictly increasing at all x’s such that 0 < F(x) < 1. Show that qy(a−X)= a−q1−y(X) for any a and y ∈ (0,1).
Show that if a r.v. X has a continuous d.f. F(x), then the distribution of the r.v. Y = F(X) coincides with the distribution uniform on [0,1]. (The r.v. Y may not assume values 0 and 1, but for the
Let two d.f.’s, F1(x) and F2(x), be such that F1(x) ≤ F2(x) for all x . This relation referred to as the first stochastic dominance (FSD) was already considered and interpreted. Show that we can
Forspecify the decomposition, the mean and variance.
For a d.f. F(x), the jumpsfor k = 1,2, .... At all other points, F(x) is differentiable, and F′(x) = a for some number a and 0 ≤ x
Using Excel or another software, simulate the process of flipping a non-regular coin with a probability of heads p as an input parameter.
Simulate 100 values of a r.v. X assuming values −1, 0, and 1, with respective probabilities 0.3, 0.4, and 0.3. Compute E{X} and compare it with the average of the numbers you simulated. Did you get
Using Excel or another software, simulate 5 values of (a) The Γ-r.v. from Example 4.3-1, (b) a r.v. X having the d.f. F(x) = x3/8 for x ∈ [0.2].
Suggest simulation methods for(a) The Binomial distribution, and(b) The geometric distribution, proceeding from the independent-trials model.
Estimate by the Monte Carlo method ∫20 x3dx and compare the result with the precise value of this integral.
Find a discrepancy between the histogram in Fig. 14 and the exponential density.
Simulate 1000 values of a r.v. with the d.f. F(x) = √x on [0,1], build a histogram, and compare it with the real density.
Provide an Excel sheet or write a program using another software to simulate a sequence of values of Poisson r.v., where λ is an input parameter.
Find the m.g.f. of a r.v. assuming values −1 and 1 with equal probabilities, and the m.g.f. of a r.v. uniform on [−1,1]. Graph both m.g.f.’s in one picture and comment in the terms of moments
Graph the m.g.f. of an exponential r.v. with a given mean. For which z’s can we do it?
Show that for an integer ν, (1.2.2) immediately follows from (1.2.1). (Advice: Think about Nν as the moment of the νth success.)

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