Questions and Answers of Probability And Stochastic Modeling

Let X1,X2 be independent standard exponential r.v.’s. we proved that the r.v. Y = X1 −X2 has the two-sided exponential density f (x) = 1/2e−|x|. Find the m.g.f. and density of Y using the
Which functions g(z) in Fig. 3abc look as m.g.f.’s?
Show that if P(X ≥ 3) = 0.2, then the corresponding m.g.f. M(z) ≥ 0.2e3z. What can we say about the distribution of X if M(z) ∼ 0.2e3z as z→∞ ?
Let M(z) be the m.g.f. of a r.v. X. When does there exist at least one number z0 > 0 such that M(z0) < 1 ? Justify your answer.
Let M(z) be the m.g.f. of a r.v. X such that E{X} = 1. Is it true that M(z) ≥ 1 for all z ≥ 0? Answer the same question for the case E{X} = −1.
Let M(z) be the m.g.f. of a r.v. X, E{X} = m, Var{X} = σ2. Write M′(0) and M″(0).
Compare the means and variances of the r.v.’s whose m.g.f.’s are graphed in Fig. 3d.
(a) Can the function 1+z4 be the m.g.f. of a r.v.?(b) In general, show that if g(z) = 1+ε(z)z2, where ε(z) ≠ 0 for z ≠ 0, and ε(z) → 0, as z→0, then g(z) cannot be a m.g.f.
Prove that the sum of independent Poisson r.v.’s is Poisson by the method of m.g.f.’s.
We come back to Section 2.2. Show that the function fS(x) = e−x/2 −e−x for x ≥ 0 and = 0 otherwise is a probability density; that is, it is non-negative and the total integral is one.
Using the method of m.g.f.’s, find the density of the sum of two independent exponential r.v.’s with means 2 and 3, respectively.
If Sn = X1 +...+Xn, where the X’s are i.i.d., then as we know, MSn (z) = (MX (z))n, where MX (z) is the m.g.f. of Xi. Show that Proposition 2 includes this case.
The number of customers of a company for a randomly chosen day is a Poisson r.v. with parameter λ = 50. The amount spent by each customer is an exponential r.v. with a mean of m = 100. All the
Assume that, in an area, the number of traffic accidents on a randomly chosen day is a Poisson r.v. with λ = 300, and the probability that a separate accident causes serious injuries is p = 0.07.
Assume that, in an area, the number of traffic accidents on a randomly chosen day is a Poisson r.v. with parameter λ1, and the number of injuries a particular accident causes is a Poisson r.v. with
The number of customers of a company during a randomly chosen day is a r.v. K for which P(K = k) = 1/3(2/3)k, k = 0,1, .... The amount spent by a particular customer is an exponential r.v. with a
Solve the problems of Examples 2.4-1,3 (that is, find the quantities A, a, and π) for the case when the remaining lifetime T is exponential with parameter μ (we are using the symbol a for other
Write a bound of the type (3.1.2) for the standard normal distribution and compare it with the bound (6.2.4.6). Which is better? Do you find the difference significant?
John and Mike play a game consisting in tossing a coin. If the coin comes up heads John pays to Mike $1. Otherwise, it is Mike who pays $1. Let Sn is Mike’s gain (which may be negative) after n
In a poll concerning future elections, 55% of 1,500 respondents said that they would vote for Candidate A. To what degree can we expect that A will win?
On Sundays, Michael sells the Sunday issue of a local newspaper. The mean time between consecutive buyers is 3 min., and the corresponding standard deviation is 1.5 min. Estimate the time period that
A surveyor is measuring the distance between two remote objects with use of geodesic instruments from 25 different positions. The errors of different measurements are independent r.v.’s with zero
Let Sn = X1 +...+Xn, where X’s are independent and uniform on [0,1]. Consider S∗n = (Sn − n/2)/√n/12. Explain why it makes sense to consider this r.v. Using software, for n = 5, 10, 20,
A regular die is rolled n times. Let Sn be the total sum of the numbers showed up. Say without calculations what Sn is equal to on the average. For a > 0, let a+n = 3.5n+a√n, a−n = 3.5n −
Let X1,X2, ... be a sequence of independent r.v.’s uniformly distributed on the interval [0,2], and let Sn = X1 + ... + Xn.(a) Find E{Sn} and Var{Sn}.(b) Using the Central Limit Theorem, estimate
Let X1,X2, ... be a sequence of independent exponential r.v.’s with E{Xi} = 2, and let Sn = X1+...+Xn. Find limn→∞ P(Sn ≤ 2n+ √n).
An energy company provides services for a town of n=10,000 households. For a household, the size of monthly energy consumption is exponentially distributed with a mean of 800 kwh, and does not depend
Show that the negative binomial distribution with parameters (p, ν) may be well approximated by a normal distribution for large ν. State the CLT for this case.
Let Y be a standard normal r.v., and for a set B in the real line, let Φ(B) = P(Y ∈ B). From Theorem 1, we know that P(S∗n ∈ B) → Φ(B) as n → ∞ if B is an interval. Is the same true for
Show that the distribution of Zλ may be approximated by a normal distribution for any large, not necessary integer, λ. (Advice: First, λ = [λ]+{λ}, where [λ] is the integer part of λ, and
What would the probability that the company will not suffer a loss have been equal to if the premium had been equal to the mean payment?
Consider the probability that the company will not suffer a loss. Using the particular data from this example, for a portfolio of n = 1000 independent policies, find a single premium (that is, the
Proceeding from Theorems 2 and 3, explain why the Cauchy distribution (which is stable) cannot be a limiting distribution for the normalized sum of i.i.d. r.v.’s with a finite variance.
Under which conditions is (2.1.2) true if the X’s are normal?
Consider a generalization of Example 2.1-4 for an arbitrary parameter a. For which a does the CLT hold? Consider the order of the Lyapunov fraction Ln (and hence, the rate of convergence in the CLT)
Show that if (2.2.3) holds, then all terms Xi/Bn are asymptotically negligible; more precisely, that
Lindeberg’s condition (2.2.2) requires Λ(ε)→0 for any ε > 0. Can it be true for ε = 0? To what, as a matter of fact, is Λ(0) equal?
Mark each statement below “true” or “false”. Justify your answers.(a) Cov{X1,X2} =Cov{X2,X1}.(b) If two r.v.’s are independent, then they are uncorrelated.(c) If two r.v.’s are
(a) Show that for Cov{X1,X2} = E{X1X2}, it suffices that either X1 or X2 has zero mean.(b) Show that Cov{X +Y,Z} =Cov{X,Z}+Cov{Y,Z}.
(a) What is Cov{X,−X}, Cov{−X,−X})?(b) Let c =Cov{X1,X2}. Write Cov{2X1,3X2}, Cov{−2X1,3X2}, Cov{−2X1,−3X2}.(c) Let ρ = Corr{X1,X2}. Write Corr{X1,−X2}, Corr({X1,10−X2},
Let a r.vec. (X,Y) be uniformly distributed on the disk {(x,y) : x2 + y2 ≤ 4}. Find the covariance. Are X,Y independent? Do the same for the uniform distribution on {(x,y) :x2 +x+y2−3y≤ 20}.
Does it follow from the Cauchy-Schwartz inequality (Proposition 1) that for any r.v.’s ξ and η with finite second moments, (E{|ξη|})2 ≤ E{ξ2}E{η2}? Which inequality is stronger: this or
Two dice are rolled. Let X and Y be the respective numbers on the first and the second die, and Z =Y −X.(a) Do you expect negative or positive correlation between X and Z?(b) Find Corr{X,Z}.
Let r.v.’s X and Y be independent, Z1 = X +Y, Z2 = X −Y. When are the r.v.’s Z1 and Z2 uncorrelated? Would it mean that they are independent? Let, say, (X,Y) be uniformly distributed on the
Let r.v.’s X and Y be independent, Z = X +Y. Find Corr{X,Z} as a function of the ratio k =Var{Y}/Var{X}. Comment on the fact that this function is monotone.
Two balls, one at a time, are drawn without replacement from a box containing r red and b black balls.(a) For i = 1, 2, let Xi = 1 if the ith ball is red, and Xi = 0 otherwise. Do you expect
Suppose n balls are distributed at random into k boxes. Let Xi = 1 if box i is non-empty, and Xi =0 otherwise. Argue that Corr{Xi,Xj}=Corr{X1,X2} for all i = j and findCorr{X1,X2}.
Let a r.vec. (X,Y) have the joint density f(x,y)= 3/8 (x+y)2 for |x|≤ 1, |y|≤1, and f (x,y)=0 otherwise. Do you expect the r.v.’s to be dependent? Find Corr{X,Y}. (Use of software is
You are working with r.v.’s X1,X2,X3, ... , and you are interested only in their variances and correlations. Can you switch to the centered r.v.’s and deal only with them?
Prove (1.1.11).
John invests one unit of money in two business projects; an amount of α goes to the first, and 1−α goes to the second. For i = 1, 2, let Xi be the (random) return (per $1) of project i. It is
Let X1,X2,X3,X4 be independent r.v.’s with variances 1,2,3,4, respectively. Do you need to know the mean values of X1,X2,X3,X4 in order to compute the correlations of(a) X1 + X2 and
(a) Let X be uniform on [−1,1], andY = X3. Find ρ =Corr{X,Y} and explain why ρ ≠ 1 while the association (or dependence) is perfect and positive.(b) Solve the same problem for Y = X2k+1, where
Represent the “beta of security” terms of the correlation coefficient ρ =Corr{X,Y}.
In a country, the mean weight of adult females is 110 lb, and the mean height is 155 cm. The respective standard deviations are 15 and 12. Ms. K weighs 114 lb, and her height is 159 cm.The same
For which ρ does there exist the inverse of C in (2.1.1)?
Given the covariance matrix C of a r.vec. (X1, ...,Xk), how to compute the variance of the sum X1+...+Xk?
Let X be a r.vec., E{X} = 0, and Y = Q X, where Q is a non-random orthogonal matrix. Using a probability theory argument, prove that the covariance matrices CY and CX have the same traces (the sums
Let (X1, ...,Xk) be a r.vec. with a covariance matrix C. Under which condition on C does there exist a linear combination t1X1+...+tkXk with zero variance and with not all t’s equal zero?(Say, you
Let X = (X1, ...,Xk) be a random vector with a covariance matrix C, and t = (t1, ..., tk) and ˜t= (˜t1, ..., ˜tk) be two non-random vectors. Consider two linear combinations: 〈X, t〉 and
Regarding relation (2.2.2), show thatExplain why a squared loading coefficient may be viewed as the percentage of the factor variance “contributed” to the variance of the component.
This exercise concerns the theory of Section 2.3. Suppose that a financial market consists of two securities with expected returns of 6% and 8%, and standard deviations of 1% and 2%, respectively.
Let f (x1,x2) be the density of a r.vec X = (X1,X2). We call a curve in a (x1,x2)-plane a level curve if it is determined by an equation f (x1,x2) = c, where c is a constant. Argue that all points
Let X and Y be independent standard normal r.v.’s, Z1 = X +Y, Z2 = X −Y. Are Z1 and Z2 independent?
Let a r.vec (X1,X2) is normal with zero means and covariance matrix (2.1.1). Write the density of X1+X2.
Let X = (X1, X2) be a two-dimensional normal random vector; E{X1} = 1, E{X21} = 5, E{X2} = 2, E{X21} = 13, E{X1X2} = −2.(a) Write the density of the centered vector Y = X−m, where the vector m =
Let the covariance matrix of a normal r.vec.Explain without any calculations why Y3 does not depend on the r.vec. (Y1,Y2). Assuming E{Y} = 0, write the density of Y.
Let us present a two-dimensional standard normal r.vec. X in the polar coordinates as (R, Θ), where R is the length of X and Θ is the angle between X and the first axis.(a) Are the r.v.’s R, Θ
Let B be a rotation matrix, X be a standard normal r.vec., and Y = BX. What is the distribution of |Y|2?
In the framework of Section 3.2, find the density of the r.v. χk = |X|.
Show that E{χk} = √2Γ((k + 1)/2)/Γ(k/2). (Advice: A simple way is to consider a r.v. U having the χ2k -distribution; that is, the Γ-distribution with parameters (1/2 , k/2 ), and write the
Continue the calculations. (Advice: You may choose a slightly heuristic approach using the representation ξk = 1/√k Yk, where Yk = 1/√k Σki=1(X2i−1), and Xi’s are standard normal. For
John throws a dart at a circular target with a radius of r. Suppose that with respect to a system of coordinates with the origin at the center of the target, the point where the dart lands may be
In the case you solved Exercise 36, find the mean speed of a molecule. (Advice: Use the fact that Γ(t +1) = tΓ(t).)Exercise 36Show that E{χk} = √2Γ((k + 1)/2)/Γ(k/2). (Advice: A simple way is
Restate Theorem 7 for the one dimensional case. How does this restatement differ from Theorem 1 from Chapter 9? Show that both assertions are equivalent.
What is the matrix D in (3.3.2) if the matrix C in Section 3.3 is diagonal?
Suppose that the coordinates of a r.vec. X = (X1, ...,Xk) are independent. Present the m.g.f MX(z) in terms of the m.g.f.’s of the coordinates. Is MX(z) uniquely determined by the m.g.f.’s of the
In Section 3.3, let k = 2, and separate terms Xi = (Xi1,Xi2), where Xi1,Xi2 are independent and have the standard exponential distribution. Let the vector e= (1,1). For large n, estimate P(|Sn −n
Is (3.3.1) true for the sets depicted? Theorem 7 is stated for convex sets, but it does not mean that (3.3.1) cannot be true for a non-convex set.
For a husband and wife, the future lifetimes are independent continuous r.v.’s X1 and X2. Suppose the tail functions F̅1(x) = P(X1 > x) and F̅2(x) = P(X2 > x) are given. Find the probabilities
(a) In a closed auction, the buyers submit sealed bids so that no bidder knows the bids of the others. The highest bid wins. Suppose that the bids are independent r.v.’s uniformly distributed on
Each day, John is jogging the same route. The times it takes are i.i.d. r.v.’s assuming values between 30 and 31 minutes. Find the distribution of the best result in a week and its expected value
Let X1,X2,X3, ... be independent exponential r.v.’s, and let E{Xi} = 2i. Find the distribution of min{X1,X2, ...} or, more precisely, the limiting distribution of
Comment on the behavior of the expectations for large n. If you pick (or simulate) 100 independent numbers from [0,1], to what will the largest and the smallest number be close? The reader who chose
The time to failure for each of particular standard electronic devices has the density f (x) = 64/x5 for x ≥ 2. Show that the corresponding r.v.’s are larger than 2 with probability one. Let Tn
Consider the system with the configuration presented in Fig. 10a and suppose that all components work independently. Let the time to failure for a separate component have a d.f. F(x).Find the d.f. of
A system consists of two components, and for the system to work, at least one component should work. The times to failure for the components are independent and exponential with a mean of one unit of
Consider the configurations in Fig. 10b, suppose that the lifetimes of the separate components are independent, denote by G(x) the d.f. of the lifetime of the “middle” component, and suppose that
Consider i.i.d. r.v.’s X1,X2, ... assuming a finite number of values x1 2 m. (a) Describe the behavior of the r.v.’sandheuristically.(b) Provide a rigorous explanation. How “fast” do the
Prove (2.1.2).
What is α and the distribution of W̰*nin the case (2.2.5) if the X’s are exponential?
Write the limiting distribution and representation (2.2.5) for the X’s having(a) The density f(x) = 2x for x ∈ [0,1];(b) The density f(x) = 1/2(2−x) for x ∈ [0,2].
Write the limiting distribution and representation (2.2.6) for the X’s having the density f(x)= 2(x−1) for x ∈ [1,2]. Compare with Exercise 13a.Exercise 13aWrite the limiting distribution and
Write the limiting distribution and representation (2.3.4) for the following situations:(a) The density of the X’s is f (x)=2(x−1) for x ∈ [1,2];(b) The density of the X’s is f (x)=3(2−x)2
Show that if in the normalization procedure (2.2.2), we multiply W̰n by (cn)1/α, then the limiting distribution will be the Weibull distribution Q1α(x) = 1−exp{−xα}.
Regarding Propositions 1 and 2, does the increase of the parameter α lead to the faster or slower rate of convergence to the limit? Give a rigorous answer and a heuristic commonsense interpretation.
Prove that the expected value of the estimate W̃n, that is, E{W̃n} = n+1 θ. Such an estimate is called biased: its expected value is not equal to the parameter we are estimating. By what should we
Let X1, ...,Xn have Weibull’s distributions Qc1α, ...,Qcnα, respectively. Show that the min{X1, ...,Xn} has the distributionQcα with c = c1+...+cn. That is, the Weibull distribution is stable
Make sure that you understand why the graphs of and indeed look as they are depicted in Fig. 6.
Write the limiting distribution and representation (2.4.8) for the X’s having the density f(x)= 64/x5 for x ≥ 2.

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