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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

The Carter and Falconer [4] map function has inverse M−1(θ) = 1 4[tan−1(2θ) + tanh−1(2θ)].Prove that the map function satisfies the differential equation M(d)=1 − 16M4(d)with initial
Continuing Problem 4, prove that Felsenstein’s map function has inverse d = 1 2(γ − 2) ln 1 − 2θ1 − 2(γ − 1)θ.
Show that Felsenstein’s [9] map functionθ = 1 2e2(2−γ)d − 1 e2(2−γ)d − γ + 1 (12.22)arises from a stationary renewal model when 0 ≤ γ ≤ 2.Kosambi’s map function is the special
Consider a delayed renewal process generated by the sequence of independent random variables X1, X2,... such that X1 has distribution function G(x) and Xi has distribution function F(x) for i ≥
Karlin’s binomial count-location model [12] presupposes that the total number of chiasmata N has binomial distribution with generating function Q(s)=( 1 2 + s 2 )r. Compute the corresponding map
Prove that in Haldane’s model the gamete probability formula (12.6)collapses to the obvious independence formula yS = k i=1θsi i (1 − θi)1−si .
The construction of radiation hybrids always involves a selectable enzyme such as HPRT or TK. On the chromosome containing the selectable locus, at least one fragment containing the locus is
In computing the distribution of the number of obligate breaks per clone in the polyploid model, how must the initial conditions and recurrences for the probabilities pk(i, j) be modified when some
Under the polyploid model for two loci, show that the expected information for θ is Jθθ = c2r2(1 − θr)c−2(1 − r)c[1 − 2(1 − r)c + (1 − θr)c][1 − (1 − θr)c][1 − 2(1 − r)c +
Under the polyploid model for two loci, consider the map(θ, r) → (q00, q11)q00 = Pr(X1 = 0, X2 = 0)q11 = Pr(X1 = 1, X2 = 1).Show that this map from {(θ, r) : θ ∈ [0, 1], r ∈ (0, 1)} is one
Complete the calculation of the partial derivatives of the likelihood for a single clone under the polyploid model by specifying the partial derivatives ∂∂θi tc,i, ∂∂r tc,i, and
Continuing the last problem, prove that Jθiθi has a maximum at r = 1 2when θi is fixed. Use this fact to show that Jθiθi ≤ 1/[2θi(1 − θi)].Given a known retention probability r, this
Let L(γ) be the loglikelihood for the data X on a single, haploid clone fully typed at m loci. Here γ = (θ1,...,θm−1, r)t is the parameter vector. The expected information matrix J has entries
Let ˆθnjk be any strongly consistent sequence of estimators of θjk for polyploid radiation hybrid data. Prove that minimizing the estimated total distance D(σ) = −m−1 i=1 ln[1 −
In addition to the assumptions of the last problem, suppose that there are just m = 2 loci. Prove that the estimates (11.23) and (11.24)of r and θ reduce to the maximum likelihood estimates
In a haploid radiation hybrid experiment with m loci, let Xij be the observation at locus j of clone i. Assuming independence of the clones and no missing data, show that rˆn = 1 nn i=1 1m mj=1
For m loci in a haploid clone with no missing observations, the expected number of obligate breaks E[B(id)] is given by expression(11.2).(a) Under the correct order, show [1] that Var[B(id)] = 2r(1
In the Ising model, one can explicitly calculate the partition functionc eH(c) of Section 10.10. To simplify matters, we impose circular symmetry and write H(c) = θ0 mi=1 ci + θ1 mi=1 1{ci=ci+1},
In Lake’s balanced transversion model of the last problem, show thatλAGλGT = λAT λGAλCT λT A = λT CλCA are necessary and sufficient conditions for the corresponding Markov chain to be
In his method of evolutionary parsimony, Lake [14] has highlighted the balanced transversion assumption. This assumption implies the constraints λAC = λAT , λGC = λGT , λCA = λCG, and λT A =
There is an explicit formula for the equilibrium distribution of a continuous-time Markov chain in terms of weighted in-trees [20]. To describe this formula, we first define a directed graph on the
For the nucleotide substitution model of Section 10.5, verify in general that the equilibrium distribution isπA = (δ + κ) + δ(γ + λ)(α + γ + + λ)(γ + δ + κ + λ)πG = α(δ + κ) + κ(γ
For the nucleotide substitution model of Section 10.5, show that Λhas eigenvalues 0, −(γ+λ+δ+κ), −(α++γ+λ), and −(δ+κ+β+σ)and corresponding right eigenvectors 1 =1, 1, 1, 1 tu
For the nucleotide substitution model of Section 10.5, prove formally that P(t) has the same pattern for equality of entries as Λ. For example, pAC(t) = pGC(t). (Hint: Prove by induction that Λk
Prove that det(eA) = etr(A), where tr is the trace function. (Hint:Since the diagonalizable matrices are dense in the set of matrices[11], by continuity you may assume that A is diagonalizable.)
Define matrices A = a 0 1 a, B = b 1 0 b.Show that AB = BA and that eAeB = ea+b 1 1 1 2 eA+B = ea+bcosh(1) 1 0 0 1 + sinh(1) 0 1 1 0 .Hence, eAeB = eA+B. (Hint: Use Problem 10 to
Let A and B be the 2 × 2 real matrices A = a −b b a , B = λ 0 1 λ.Show that eA = ea cos b − sin b sin b cos b, eB = eλ 1 0 1 1 .(Hints: Note that 2 × 2 matrices of the form a −b
Let P(t)=[pij (t)] be the finite-time transition matrix of a finite-state irreducible Markov chain. Show that pij (t) > 0 for all i, j, and t > 0.Thus, no state in a continuous-time chain displays
Let Λ be the infinitesimal transition matrix of a Markov chain, and suppose µ ≥ maxi λi. If R = I + 1µΛ, prove that R has nonnegative entries and that S(t) = ∞i=0 e−µt (µt)i i! Ri
Let Λ = (Λij ) be an m × m matrix and π = (πi) be a 1 × m row vector. Show that the equality πiΛij = πjΛji is true for all pairs(i, j) if and only if diag(π)Λ = Λt diag(π), where
Let Λ be the infinitesimal transition matrix and π the equilibrium distribution of a reversible Markov chain with n states. Define an inner product u, vπ on complex column vectors u and v with n
Consider a continuous-time Markov chain with infinitesimal transition matrix Λ = (Λij ) and equilibrium distribution π. If the chain is at equilibrium at time 0, then show that it experiences ti
Let u and v be column vectors with the same number of components.Applying the definition of the matrix exponential (10.7), show that esuvt=, I + suvt if vtu = 0 I + esvt u−1 vtu uvt otherwise Using
Consider four contemporary taxa numbered 1, 2, 3, and 4.A total of n shared DNA sites are sequenced for each taxon. Let Nwxyz be the number of sites at which taxon 1 has base w, taxon 2 base x, and
In the notation of Section 10.2, let Sn = T2 + ··· + Tn. Prove the inequalities Tn1 + n − 2 2n2≤ Sn ≤ Tn1 +1 n − 1for all n ≥ 2.
Compute the number of unrooted evolutionary trees possible for n contemporary taxa. (Hint: How does this relate to the number of rooted trees?)
One can adapt the Lander-Green-Kruglyak algorithm to perform haplotyping. In the notation of Section 9.11, define γi(yi | j) to be the likelihood of the most likely descent state consistent with the
Consider the set G of column vectors j = (j1,...,jm)t whose entries are either 0 or 1.If m = 3, a typical vector in G is (0, 1, 1). Altogether G has 2m elements. Prove that G forms a commutative
Formally describe the transition rules T2a and T2b on descent graphs in terms of the transition rules T0 and T1.
Another device to improve mixing of a Markov chain is to run several parallel chains on the same state space and occasionally swap their states [9]. If π is the distribution of the chain we wish to
Importance sampling is one remedy when the states of a Markov chain communicate poorly [13]. Suppose that π is the equilibrium distribution of the chain. If we sample from a chain whose distribution
If the component updated in Gibbs sampling depends probabilistically on the current state of the chain, how must the HastingsMetropolis acceptance probability be modified to preserve detailed
The Metropolis acceptance mechanism (9.6) ordinarily implies aperiodicity of the underlying Markov chain. Show that if the proposal distribution is symmetric and if some state i has a neighboring
Let Z0, Z1, Z2,... be a realization of an ergodic chain. If we sample every kth epoch, then show (a) that the sampled chain Z0, Zk, Z2k,...is ergodic, (b) that it possesses the same equilibrium
For an irreducible chain, demonstrate that aperiodicity is a necessary and sufficient condition for some power P n of the transition matrix P to have all entries positive. (Hint: For sufficiency, you
Find a transition matrix P such that limn→∞ P n does not exist.
“Selfing” is a plant breeding scheme that mates an organism with itself, selects one of the progeny randomly and mates it with itself, and so forth from generation to generation. Suppose at some
The restriction enzyme HhaI has the recognition site GCGC. Formulate a Markov chain for the attainment of this restriction site when moving along a DNA strand. What are the states and what are the
Numerically find the equilibrium distribution of the Markov chain corresponding to the AluI restriction site model. Is this chain reversible?
Continuing Problem 14, let vm be the trait variance of a person m generations removed from his or her relevant pedigree founders in a non-inbred pedigree. Verify that vm satisfies the difference
In the hypergeometric polygenic model, suppose that one randomly samples each of the n polygenes transmitted to a gamete with replacement rather than without replacement. If j = i is not a
In the hypergeometric polygenic model, Var(Xi)=2n holds for each person i in a non-inbred pedigree. In the presence of inbreeding, give a counterexample to this formula. However, prove that 0 ≤
In the hypergeometric polygenic model, verify that the number of positive polygenes a non-inbred person possesses follows the binomial distribution (8.12). Do this by a qualitative argument and by
Any reasonable model of QTL mapping for an X-linked trait must take into account the phenomenon of X inactivation in females. As a first approach, assume that all females are divided into n patches
In the factor analysis model of Section 8.7, we can exploit the approximate multivariate normality of the estimators to derive a different approximation to the parameter asymptotic standard errors.
In some variance component models, several pedigrees share the same theoretical mean vector µ and variance matrix Ω. Maximum likelihood computations can be accelerated by taking advantage of this
Demonstrate the following facts about the Kronecker product of two matrices:(a) c(A ⊗ B)=(cA) ⊗ B = A ⊗ (cB) for any scalar c.(b) (A ⊗ B)t = At ⊗ Bt.(c) (A + B) ⊗ C = A ⊗ C + B ⊗
Continuing Problem 5, suppose that one or more of the covariance matrices Γk is singular. For instance, in modeling common household effects, the corresponding missing data Xk can be represented as
Continuing Problem 5, show that σ2 nk ≥ 0 holds for all k and n ifσ2 1k ≥ 0 holds initially for all k. If all σ2 1k > 0, show that all σ2 nk > 0.
As an alternative to scoring in the polygenic model, one can implement the EM algorithm [7]. In the notation of the text, consider a multivariate normal random vector Y with mean ν = Aµ and
Suppose all pedigrees from a sample have been amalgamated into a single pedigree. For a trait vector Y with E(Y ) = 0, consider the covariance components model YiYj = r k=1σ2 kΓkij + eij ,
Verify that the formulas (8.3) for the expected information matrix continue to hold when the mean A(µ) and the covariance Ω(σ2) are nonlinear functions of the underlying parameter vectors µ and
(Hint:What is the distribution of Ω− 1 2i (Y i−Aiµ)? Recall that a linear transformation of a multivariate normal variate is multivariate normal.)
In the notation of Problem 1, prove that the pedigree statistic(Y i − Aiµ)tΩ−1 i (Y i − Aiµ)has a χ2 mi distribution when evaluated at the true values of µ and σ2[30]. This χ2 mi
Suppose that Aiµˆ and Ωˆi are the mean vector and covariance matrix for the ith of s pedigrees evaluated at the maximum likelihood estimates. Under the multivariate normal model (8.1), show that
Consider the revision L(β) =G1···Gni Pen(Xi | Gi)β1{Gi=g}×j Prior(Gj ) {k,l,m}Tran(Gm | Gk, Gl)of the likelihood expression (7.1), where β is an artificial parameter and g is a fixed
Consider a nuclear family in which one parent is affected by an autosomal dominant disease [11]. If the affected parent is heterozygous at a codominant marker locus, the normal parent is homozygous
The grandson 7 depicted in the pedigree of Figure 7.8 is afflicted by a lethal, X-linked recessive disease [25]. Problem 11 of Chapter 1 notes that if the carrier females for such a disease are fully
A healthy male had a sister with cystic fibrosis (CF), but she and his parents are dead. What is his risk of being a carrier for this recessive disease? About 75 percent of all disease alleles at the
Figure 7.7 gives a pedigree for an autosomal recessive disease and a linked marker. The four marker genesa, b,c, and d are assumed distinct. If the recombination fraction between the two loci is θ,
Do by hand the risk prediction calculation for myotonic dystrophy, showing the various steps of the computations in detail. Neglect the extremely rare Dm +/Dm+ genotype at the myotonic dystrophy
Show that the greedy tactic of assembling the product array from the pairwise products of the multiplicand arrays first multiplies B times D, then A times C, and finally the product BD times the
Consider the array product E(G1, G2, G3, G4, G5)= A(G1, G2, G3)B(G1)C(G2, G3, G4)D(G5), where the range set Si for the index Gi has 4, 2, or 3 elements according as i = 1, i ∈ {2, 3, 4}, or i =
The sum of array productsG1∈S1···G9∈S9 A(G1, G2, G3, G4)B(G4, G5)× C(G5, G6)D(G6, G7, G8, G9)can be evaluated as an iterated sum by the greedy algorithm. If all range sets Si have the same
Consider the partially typed, inbred pedigree depicted in Figure 7.6.The phenotypes displayed in the figure are unordered genotypes at a single codominant locus with three alleles. Show that the
Under Haldane’s model of independent recombination on disjoint intervals, it is possible to compute the recombination fraction θij between two loci i
In the pedigree depicted , compute the marker-sharing statistic Z and its expectation E(Z) for the three phenotyped affecteds 3, 4, and 6.Assume f(p)=1/p, pa = 1/2, pb = 1/4, and for a third
In the two-locus heterogeneity model with X = Y + Z − Y Z, carry through the computations retaining the product term Y Z. In particular, let Km be the prevalence of the mth form of the disease, and
Let (X1,...,Xn) and (Y1,...,Yn) be measured values for two different traits on a pedigree of n people. Suppose that both traits are determined by the same locus. Show that there exist constants σaxy
Show that the matrices Φ and ∆7 of coefficients assigned to a pedigree do not necessarily commute. It is therefore pointless to attempt a simultaneous diagonalization of these two matrices. (Hint:
Prove that any pair of nonnegative numbers (σ2a, σ2d) can be realized as additive and dominance genetic variances. The special pairs ( 1 2 , 0)and (0, 1) show that the two matrices Φ = (Φij ) and
For a locus with two alleles, show that the additive genetic variance satisfiesσ2 a = 2p1p2(α1 − α2)2= 2p1p2[p1(µ11 − µ12) + p2(µ12 − µ22)]2. (6.10)As a consequence of formula (6.10),
Suppose that the two relatives i and j are inbred. Show that the covariance between their trait values Xi and Xj is Cov(Xi, Xj ) = (4∆1 + 2∆3 + 2∆5 + 2∆7 + ∆8)kα2 kpk+ (4∆1 + ∆3 +
Suppose that marker loci on different chromosomes are typed on two putative relatives. At locus i, let pij be the likelihood of the observed pair of phenotypes conditional on the relatives being in
Consider a disease trait partially determined by an autosomal locus with two alleles 1 and 2 having frequencies p1 and p2. Let φk/l be the probability that a person with genotype k/l manifests the
Let the disease allele at a recessive disease locus have population frequency q. If a child has inbreeding coefficientf, argue that his or her disease risk is fq + (1 − f)q2. What assumptions does
The definition of a generalized X-linked kinship coefficient exactly parallels the definition of a generalized kinship coefficient except that genes are sampled from a generic X-linked locus rather
Wright proposed a path formula for computing inbreeding coefficients that can be generalized to computing kinship coefficients [15]. The pedigree formula isΦij =pij1 2n(pij )[1 + fa(pij )], where
Geneticists employ repeated sib mating to produce inbred lines of laboratory animals such as mice. At generation 0, two unrelated animals are mated to produce generation 1.A brother and sister of
Selfing is a mating system used extensively in plant breeding. As its name implies, a plant is mated to itself, then one of its offspring is mated to itself, and so forth. Let fn be the inbreeding
We define the X-linked kinship coefficient Φij between two relatives i and j as the probability that a gene drawn randomly from an Xlinked locus of i is i.b.d. to a gene drawn randomly from the same
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
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,
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
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
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,
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
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
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,
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

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