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probability and stochastic modeling

An Introduction To Stochastic Modeling 3rd Edition Samuel Karlin, Howard M. Taylor - Solutions

Continuing Problem 5, we can extract some of the eigenvectors and eigenvalues of a kinship matrix of a general pedigree [16]. Consider a set of individuals in the pedigree possessing the same inbreeding coefficient and the same kinship coefficients with other pedigree members.Typical cases are a
Explicit diagonalization of the kinship matrix Φ of a pedigree is an unsolved problem in general. In this problem we consider the special case of a nuclear family with n siblings. For convenience, number the parents 1 and 2 and the siblings 3,...,n+2. Let ei be the vector with 1 in position i and
The Cholesky decomposition of a positive definite matrix Ω is the unique lower triangular matrix L = (lij ) satisfying Ω = LLt and lii > 0 for all i. Let Φ be the kinship matrix of a pedigree with n people numbered so that parents precede their children. The Cholesky decomposition L of Φ can
Continuing Problem 5, we can extract some of the eigenvectors and eigenvalues of a kinship matrix of a general pedigree [16]. Consider a set of individuals in the pedigree possessing the same inbreeding coefficient and the same kinship coefficients with other pedigree members.Typical cases are a
Explicit diagonalization of the kinship matrix Φ of a pedigree is an unsolved problem in general. In this problem we consider the special case of a nuclear family with n siblings. For convenience, number the parents 1 and 2 and the siblings 3,...,n+2. Let ei be the vector with 1 in position i and
The Cholesky decomposition of a positive definite matrix Ω is the unique lower triangular matrix L = (lij ) satisfying Ω = LLt and lii > 0 for all i. Let Φ be the kinship matrix of a pedigree with n people numbered so that parents precede their children. The Cholesky decomposition L of Φ can
Continuing Problem 5, we can extract some of the eigenvectors and eigenvalues of a kinship matrix of a general pedigree [16]. Consider a set of individuals in the pedigree possessing the same inbreeding coefficient and the same kinship coefficients with other pedigree members.Typical cases are a
Explicit diagonalization of the kinship matrix Φ of a pedigree is an unsolved problem in general. In this problem we consider the special case of a nuclear family with n siblings. For convenience, number the parents 1 and 2 and the siblings 3,...,n+2. Let ei be the vector with 1 in position i and
The Cholesky decomposition of a positive definite matrix Ω is the unique lower triangular matrix L = (lij ) satisfying Ω = LLt and lii > 0 for all i. Let Φ be the kinship matrix of a pedigree with n people numbered so that parents precede their children. The Cholesky decomposition L of Φ can
Continuing Problem 5, we can extract some of the eigenvectors and eigenvalues of a kinship matrix of a general pedigree [16]. Consider a set of individuals in the pedigree possessing the same inbreeding coefficient and the same kinship coefficients with other pedigree members.Typical cases are a
Explicit diagonalization of the kinship matrix Φ of a pedigree is an unsolved problem in general. In this problem we consider the special case of a nuclear family with n siblings. For convenience, number the parents 1 and 2 and the siblings 3,...,n+2. Let ei be the vector with 1 in position i and
The Cholesky decomposition of a positive definite matrix Ω is the unique lower triangular matrix L = (lij ) satisfying Ω = LLt and lii > 0 for all i. Let Φ be the kinship matrix of a pedigree with n people numbered so that parents precede their children. The Cholesky decomposition L of Φ can
Calculate all nine condensed identity coefficients for the two inbred siblings 5 and 6 of Figure 5.1.
Given the assumptions and notation of Problem 1 above, show that 4∆7∆9 ≤ ∆2 8 [15]. This inequality puts an additional constraint on∆7, ∆8, and ∆9 besides the obvious nonnegativity requirements and the sum requirement ∆7 + ∆8 + ∆9 = 1.(Hints: Note first thatΦij = 1 2∆7 +1
Consider two non-inbred relatives i and j with parents k and l and m and n, respectively. Show that∆7 = ΦkmΦln + ΦknΦlm∆8 = 4Φij − 2∆7∆9 = 1 − ∆7 − ∆8, where the condensed identity coefficients all pertain to the pair i and j. Thus, in the absence of inbreeding, all nonzero
Describe and program a permutation version of the two-sample t-test.Compare it on actual data to the standard two-sample t-test.
Describe and program an efficient algorithm for generating random permutations of the set {1,...,n}. How many calls of a random number generator are involved? How many interchanges of two numbers?You might wish to compare your results to the algorithm in [29].
A geneticist phenotypes n unrelated people at each of m loci with codominant alleles and records a vector i = (i1/i∗1,...,im/i∗m) of genotypes for each person. Because phase is unknown, i cannot be resolved into two haplotypes. The data gathered can be summarized by the number of people ni
Alternatively, write c1j as a sum of indicator random variables and calculate the mean and variance directly. Check that the two methods give the same answer. (Hints: In applying Problem 8, i has two components. Set all but one of the li equal to 0.Set the remaining one equal to 1 or 2 to get
Verify the mean and variance expressions in equation (4.6) using Problem
To compute moments under the Fisher-Yates distribution (4.4), let ur =u(u − 1)···(u − r + 1) r > 0 1 r = 0 be a falling factorial power, and let {li} be a collection of nonnegative integers indexed by the haplotypes i = (i1 ...,im). Setting l = i li and ljk = i 1{ij=k}li, show that Ei
The nonparametric linkage test of de Vries et al. [10] uses affected sibling data. Consider a nuclear family with s affected sibs and a heterozygous parent with genotype a/b at some marker locus. Let na and nb count the number of affected sibs receiving the a and b alleles, respectively, from the
Continuing Problem 5, define the statistic Ud to be the number of categories i with Ni
Consider a multinomial model with m categories, n trials, and probability pi attached to category i. Express the distribution function of the maximum number of counts maxi Ni observed in any category in terms of the distribution functions of the Wd. How can the algorithm for computing the
Using the Chen-Stein method and probabilistic coupling, Barbour et al. [4] show that the statistic Wd satisfies the inequality sup A⊂N|Pr(Wd ∈ A) − Pr(Z ∈ A)| ≤ 1 − e−λλ [λ − Var(Wd)],(4.9)where Z is a Poisson random variable having the same expectationλ = m i=1 µi as Wd, and
Let (N1,...,Nm) be the outcome vector for a multinomial experiment with n trials and m categories. Prove that Pr(N1 ≤ t1,...,Nm ≤ tm) ≤ m i=1 Pr(Ni ≤ ti) (4.7)Pr(N1 ≥ t1,...,Nm ≥ tm) ≤ m i=1 Pr(Ni ≥ ti) (4.8)for all integers t1,...,tm. If all tk = 0 in (4.8) except for ti and tj ,
Table 4.4 lists frequencies of coat colors among cats in Singapore [35].Assuming an X-linked locus with two alleles, estimate the two allele frequencies by gene counting. Test for Hardy-Weinberg equilibrium using a likelihood ratio test.TABLE 4.4. Coat Colors among Singapore Cats Females Males Dark
Test for Hardy-Weinberg equilibrium in the MN Syrian data presented in Chapter 2.
Problem 5 of Chapter 2 considers haplotype frequency estimation for two linked, biallelic loci. The EM algorithm discussed there relies on the allele-counting estimates pA, pa, pB, and pb.(a) Construct the Dirichlet prior from these estimates mentioned in Section 3.8 and devise an EM algorithm that
Suppose n unrelated people are sampled at a codominant locus with k alleles. If Ni = ni genes of allele type i are counted, and if a Dirichlet prior is assumed with parameters α1,...,αk, then we have seen that the allele frequency vector p = (p1,...,pk)t has a posterior Dirichlet
In the notation of Problem 14, find the score and expected information of a single observation from the Dirichlet distribution. (Hint:In calculating the expected information, take the expectation of the observed information rather than the covariance matrix of the score.)
Let Y = (Y1,...,Yk)t follow a Dirichlet distribution with parametersα1,...,αk. Compute Var(Yi) and Cov(Yi, Yj ) for i = j. Also show that (Y1 + Y2, Y3,...,Yk)t has a Dirichlet distribution.
As an application of Problems 10, 11 and 12, consider the estimation of haplotype frequencies from a random sample of people who are genotyped at the same linked, codominant loci. The resulting multilocus genotypes lack phase. Find an explicit upper bound on the expected information matrix for the
In the setting of the EM algorithm, suppose that Y is the observed data and X is the complete data. Let Y and X have expected information matrices J(θ) and I(θ), respectively. Prove that I(θ) J(θ)in the notation of Problem 9.If we could redesign our experiment so that X is observed directly,
Let X = (X1,...,Xm)t follow a multinomial distribution with n trials and m categories. If the success probability for category i is θi for 1 ≤ i ≤ m − 1 and 1 − m−1 j=1 θj for i = m, then show that X has expected information
In the notation of Problem 9, demonstrate that two positive definite matrices A = (aij ) and B = (bij ) satisfy AB if and only they satisfy B−1 A−1. If AB, then prove that det A ≥ det B, tr A ≥ tr B, and aii ≥ bii for all i. (Hints: AB is equivalent to xt Ax ≥ xtBx for all vectors
For symmetric matrices A and B, define A 0 to mean that A is nonnegative definite and AB to mean that A − B 0.Show that AB and BC imply AC. Also show that AB and BA imply A = B. Thus, induces a partial order on the set of symmetric matrices.
In the Gauss-Newton algorithm (3.19), the matrix mi=1 widµi(θn)t dµi(θn)
Let the m independent random variables X1,...,Xm be normally distributed with means µi(θ) and variances σ2/wi, where the wi are known constants. From observed values X1 = x1,...,Xm = xm, one can estimate the mean parameters θ and the variance parameter σ2 simultaneously by the scoring
A family of discrete density functions pn(θ) defined on {0, 1,...} and indexed by a parameter θ > 0 is said to be a power-series family if for all n pn(θ) = cnθn g(θ), (3.18)where cn ≥ 0 and where g(θ) = ∞k=0 ckθk is the appropriate normalizing constant. If X1,...,Xm is a random sample
Verify the loglikelihood, score, and expected information entries in Table 3.1 for the binomial, Poisson, and exponential families
Show that Newton’s method converges in one iteration to the maximum of the quadratic function L(θ) = d + etθ +1 2θt F θif the symmetric matrix F is negative definite.
Apply the result of Problem 1 to show that the loglikelihood of the observed data in the ABO example of Chapter 2 is strictly concave and therefore possesses a single global maximum. Why does the maximum occur on the interior of the feasible region?
Let f(x) be a real-valued function whose Hessian matrix ( ∂2∂xi∂xj f)is positive definite throughout some convex open set U of Rm. For u = 0 and x ∈ U, show that the function t → f(x + tu) of the real variable t is strictly convex on {t : x + tu ∈ U}. Use this fact to demonstrate that
A palindromic DNA string such as ggatcc equals its reverse complement. Amend the EM algorithm of Section so that it handles palindromic binding domain patterns. What restrictions does this imply on the domain probabilities p(m, b)?
Suppose light bulbs have an exponential lifetime with mean θ. Two experiments are conducted. In the first, the lifetimes y1,...,yn of n independent bulbs are observed. In the second, p independent bulbs are observed to burn out before time t, and q independent bulbs are observed to burn out after
In the spirit of Problem 10, formulate models for hidden Poisson and exponential trials [16]. If the number of trials is N and the mean per trial is θ, then show that the EM update in the Poisson case isθn+1 = θn +θn E(N | Y,θn)d dθL(θn)and in the exponential case isθn+1 = θn +θ2 nE(N |
Chun Li has derived an extension of Problem 10 for hidden multinomial trials. Let N denote the number of hidden trials, θi the probability of outcome i of k possible outcomes, and L(θ) the loglikelihood of the observed data Y . Derive the EM updateθn+1 i = θn i +θn iE(N | Y,θn)∂∂θi
As an example of the hidden binomial trials theory sketched in Problem 10, consider a random sample of twin pairs. Let u of these pairs consist of male pairs, v consist of female pairs, and w consist of opposite sex pairs. A simple model to explain these data involves a random Bernoulli choice for
Suppose that the complete data in the EM algorithm involve N binomial trials with success probability θ per trial. Here N can be random or fixed. If M trials result in success, then the complete data likelihood can be written as θM (1 − θ)N−Mc, where c is an irrelevant constant. The E step
In the EM algorithm, demonstrate the identity∂∂θi Q(θ | θn)|θ=θn = ∂∂θi L(θn)for any component θi of θ at any interior point θn of the parameter domain. Here L(θ) is the loglikelihood of the observed data Y .
Consider the data from the London Times [15] for the years 1910 to 1912 reproduced in Table 2.6. The two columns labeled “Deaths i”refer to the number of deaths of women 80 years and older reported by day. The columns labeled “Frequency ni” refer to the number of days with i deaths. A
In an inbred population, the inbreeding coefficient f is the probability that two genes of a random person at some locus are both copies of the same ancestral gene. Assume that there are k codominant alleles and that pi is the frequency of allele Ai. Show that f pi + (1 − f)p2 iis the frequency
In a genetic linkage experiment, AB/ab animals are crossed to measure the recombination fraction θ between two loci with alleles A and a at the first locus and alleles B and b at the second locus. In this design the dominant alleles A and B are in the coupling phase. Verify that the offspring of
Consider two loci in Hardy-Weinberg equilibrium, but possibly not in linkage equilibrium. Devise an EM algorithm for estimating the gamete frequencies pAB, pAb, paB, and pab, where A and a are the two alleles at the first locus and B and b are the two alleles at the second locus [17]. In a random
In forensic applications of DNA fingerprinting, match probabilities p2 i for homozygotes and 2pipj for heterozygotes are computed [1]. In practice, the frequencies pi can only be estimated. Assuming codominant alleles and the estimates ˆpi = ni/(2n) given in the previous problem, show that the
Consider a codominant, autosomal locus with k alleles. In a random sample of n people, let ni be the number of genes of allele i. Show that the gene-counting estimates ˆpi = ni/(2n) are maximum likelihood estimates.
Color blindness is an X-linked recessive trait. Suppose that in a random sample there are fB normal females, fb color-blind females, mB normal males, and mb color-blind males. If n = 2fB + 2fb +mB +mb is the number of genes in the sample, then show that under HardyWeinberg equilibrium the maximum
At some autosomal locus with two alleles R and r, let R be dominant to r. Suppose a random sample of n people contains nr people with the recessive genotype r/r. Prove that nr/n is the maximum likelihood estimate of the frequency of allele r under Hardy-Weinberg equilibrium
To explore the impact of genetic screening for carriers, consider a lethal recessive disease with two alleles, the normal allele A2 and the recessive disease allele A1. Mutation from A2 to A1 takes place at rate µ. No backmutation is permitted. An entire population is screened for carriers. If a
Let f(p) be a continuously differentiable map from the interval [a, b]into itself, and let p∞ = f(p∞) be an equilibrium (fixed) point of the iteration scheme pn+1 = f(pn). If |f(p∞)| < 1, then show that p∞is a locally stable equilibrium in the sense that limn→∞ pn = p∞ for p0
In the selection model of Section 1.5, it of some interest to determine the number of generations n it takes for allele A1 to go from frequency p0 to frequency pn. This is a rather difficult problem to treat in the context of difference equations. However, for slow selection, considerable progress
Consider a model for the mutation-selection balance at an X-linked locus. Let normal females and males have fitness 1, carrier females fitness tx, and affected males fitness ty. Also, let the mutation rate from the normal allele A2 to the disease allele A1 be µ in both sexes.It is possible to
Consider an autosomal dominant disease in a stationary population.If the fitness of normal A2/A2 people to the fitness of affected A1/A2 people is in the ratio 1 − s : 1, then show that the average number of people ultimately affected by a new mutation is 1−s−s . (Hints:An A2/A2 person has on
To verify convergence to linkage equilibrium for a pair of X-linked loci A and B, define Pnx(AiBj ) and Pny(AiBj ) to be the frequencies of the AiBj haplotype at generation n in females and males, respectively. For the sake of simplicity, assume that both loci are in HardyWeinberg equilibrium and
Consulting Problems 5 and 6, formulate a Moran model for approach to linkage equilibrium at two loci. In the context of this model, show that Pt(AiBj ) = e−θtP0(AiBj ) + (1 − e−θt)piqj , where time t is measured continuously
Consider three loci A—B—C along a chromosome. To model convergence to linkage equilibrium at these loci, select alleles Ai, Bj , and Ck and denote their population frequencies by pi, qj , and rk. Let θAB be the probability of recombination between loci A and B but not between B and C. Define
Consider an X-linked version of the Moran model in the previous problem. Again let u(t), v(t), and w(t) be the frequencies of the three female genotypes A1/A1, A1/A2, and A2/A2, respectively. Let r(t)and s(t) be the frequencies of the male genotypes A1 and A2(a) Verify the differential equations r
Moran [12] has proposed a model for the approach of allele frequencies to Hardy-Weinberg equilibrium that permits generations to overlap.Let u(t), v(t), and w(t) be the relative proportions of the genotypes A1/A1, A1/A2, and A2/A2 at time t. Assume that in the small time interval (t, t+dt) a
In forensic applications of genetics, loci with high exclusion probabilities are typed. For a codominant locus with n alleles, show that the probability of two random people having different genotypes is e =n−1 i=1 nj=i+1 2pipj (1 − 2pipj ) + n i=1 p2 i (1 − p2 i )under Hardy-Weinberg
Consider an autosomal locus with m alleles in Hardy-Weinberg equilibrium. If allele Ai has frequency pi, then show that a random noninbred person is heterozygous with probability 1 − m i=1 p2 i . What is the maximum of this probability, and for what allele frequencies is this maximum attained?
Suppose that in the Hardy-Weinberg model for an autosomal locus the genotype frequencies for the two sexes differ. What is the ultimate frequency of a given allele? How long does it take genotype frequencies to stabilize at their Hardy-Weinberg values?
In blood transfusions, compatibility at the ABO and Rh loci is important. These autosomal loci are unlinked. At the Rh locus, the +allele codes for the presence of a red cell antigen and therefore is dominant to the − allele, which codes for the absence of the antigen.Suppose that the frequencies
5.5. Consider a stochastic process that evolves according to the following laws: If X,, = 0, then X,,+, = 0, whereas if X,, > 0, then(a) Show that X,, is a nonnegative martingale.(b) Suppose that X0 = i > 0. Use the maximal inequality to bound Pr{X ? N for some n ? OIXO = i}.Note: X,,
5.4. Let 62, ... be independent Bernoulli random variables with parameter p, 0 < p < 1. Show that X. = 1 and X,, = p" 6, ... ,,, n = 1, 2, . . . , defines a nonnegative martingale. What is the limit of X,, as n-oo?
5.3 Let So = 0, and for n ? 1, let S,, = s, + + s be the sum of n independent random variables, each exponentially distributed with mean E[e] = 1. Show thatdefines a martingale. X 2" exp(-S), n 0
5.2. Let U,, U2, . . . be independent random variables each uniformly distributed over the interval (0, 1]. Show that X0 = 1 and X,, = 2"U, U for n = 1, 2.... defines a martingale.
5.1. Use the law of total probability for conditional expectations E[E{XIY, Z}IZ] = E[XIZ] to show E[X+2|X,,X] = E[E{X+2|X,,X+1}|X, ..., X]. n+ Conclude that when X, is a martingale. E[X+2XX] = X.
5.3. Let 6 be a random variable with mean μ and standard deviation a-.Let X = (6 - μ)Z. Apply Markov's inequality to X to deduce Chebyshev's inequality: Pr{ } for any > 0.
5.2. Let X be a Bernoulli random variable with parameter p. Compare Pr{X ? 1) with the Markov inequality bound.
5.1. Let X be an exponentially distributed random variable with mean E[X] = 1. For x = 0.5, 1, and 2, compare Pr{X > x} with the Markov inequality bound E[X]/x.
4.8. Let X and Y have the joint normal density given in I, (4.16). Show that the conditional density function for X, given that Y = y, is normal with moments and . .. + -(y - My) = - .
4.7. Suppose that X and Y are independent random variables, each having the same exponential distribution with parametera. What is the conditional probability density function for X, give that Z = X + Y: = z?
4.6. Let Xo, x, X2, . . . be independent identically distributed nonnegative random variables having a continuous distribution. Let N be the first index k for which Xk > Xo. That is, N = 1 if X, > Xo, N = 2 if X, X0 and X2 > X0, etc. Determine the probability mass function for N and the mean E[N].
4.5. Let X and Y be jointly distributed random variables whose joint probability mass function is given in the following table:Show that the covariance between X and Y is zero even though X and Y are not independent. x * -1 0 0 1 y -1 0 1 0 41 6171819 0 913 210 0 = p(x, y) Pr{Xx, Y = y}
4.4. Suppose X and Y are independent random variables having the same Poisson distribution with parameter A, but where A is also random, being exponentially distributed with parameter 0. What is the conditional distribution for X given that X + Y = n?
4.3. Let X have a Poisson distribution with parameter A > 0. Suppose A itself is random, following an exponential density with parameter 0.(a) What is the marginal distribution of X?(b) Determine the conditional density for A given X = k.
4.2. Let N have a Poisson distribution with parameter A > 0. Suppose that, conditioned on N = n, the random variable X is binomially distributed with parameters N = n and p. Set Y = N - X. Show that X and Y have Poisson distributions with respective parameters Ap and A(1 - p)and that X and Y are
4.1. Suppose that the outcome X of a certain chance mechanism depends on a parameter p according to Pr{X = 1) = p and Pr{X = 0} =1 - p, where 0 < p < 1. Suppose that p is chosen at random, uniformly distributed over the unit interval [0, 1], and then that two independent outcomes X, and X, are
4.5. Let U be uniformly distributed over the interval [0, L] where L follows the gamma density fL(x) = xe for x >_ 0. What is the joint density function of U and V = L - U?
4.4. Suppose X and Y are independent random variables, each exponentially distributed with parameter A. Determine the probability density function for Z = X/Y.
4.3. A random variable T is selected that is uniformly distributed over the interval (0, 1]. Then a second random variable U is chosen, uniformly distributed on the interval (0, T]. What is the probability that U exceeds 1?
4.2. Suppose that three components in a certain system each function with probability p and fail with probability 1 - p, each component operating or failing independently of the others. But the system is in a random environment, so that p is itself a random variable. Suppose that p is uniformly
4.1. Suppose that three contestants on a quiz show are each given the same question, and that each answers it correctly, independently of the others, with probability p. But the difficulty of the question is itself a random variable, so let us suppose, for the sake of illustration, that p is
For each given p, let X have a binomial distribution with parameters p and N. Suppose that p is uniformly distributed on the interval[0, 1]. What is the resulting distribution of X?
3.5. To form a slightly different random sum, let &, ,, . . . be independent identically distributed random variables and let N be a nonnegative integer-valued random variable, independent of , .... The first two moments areDetermine the mean and variance of the random sum E[] =, E[N] = v,
3.4. Suppose 6 62, ... are independent and identically distributed random variables having mean μ and variance 0-2. Form the random sum SN=6i +...+6N(a) Derive the mean and variance of S, when N has a Poisson distribution with parameter A.(b) Determine the mean and variance of S, when N has a
3.3. Suppose that f,, ,, ... are independent and identically distributed with Pr(6R = ± 11 = ;. Let N be independent of 6, ,2, ... and follow the geometric probability mass function p,,(k) = all - a)' for k = 0, 1, ... , where 0 < a < 1. Form the random sum Z = , + +(a) Determine the mean and
3.2. For each given p, let Z have a binomial distribution with parameters p and N. Suppose that N is itself binomially distributed with parameters q and M. Formulate Z as a random sum and show that Z has a binomial distribution with parameters pq and M.
3.1. The following experiment is performed: An observation is made of a Poisson random variable N with parameter A. Then N independent Bernoulli trials are performed, each with probability p of success. Let Z be the total number of successes observed in the N trials.(a) Formulate Z as a random sum

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