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A multivariate Gaussian distribution is the law of a Random vector $X\in {\mathbb{R}}^d$ such that every linear combination $a^{T}X$ has a Normal distribution, allowing a constant as a zero-variance Normal distribution. Its Mean vector $\mu$ and positive-semidefinite covariance matrix $\Sigma$ determine its Characteristic function:If $\Sigma$ is positive definite, the law has the normalized Probability density functionA singular covariance matrix is also permitted, but this density formula is not: the law is concentrated on the affine subspace $\mu +\mathrm{range}\Sigma$. For example, $(Z,Z)$ with $Z$ standard normal has covariance $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ and lives on the diagonal, so it has no two-dimensional Lebesgue density.

A construction valid in both cases is $X=\mu +AZ$, where $Z$ has independent standard normal coordinates and $A{A}^T=\Sigma$. Its Characteristic function follows by multiplying the one-dimensional characteristic functions. For $\mathrm{rank}\Sigma =r$, one can use a matrix $A$ with $r$ columns.

Normal marginals alone do not imply a multivariate Gaussian law. Let $Z$ be standard normal and let $S$ be an independent Rademacher random variable. Both $Z$ and $SZ$ are standard normal, but their sum is zero with probability $1/2$ and otherwise equals $2Z$, so that nonconstant sum is not normal. Joint Gaussianity is the stronger condition needed for Gaussian moment and linear-transformation rules.

Let a segment of fixed length $b$ make angle $\theta$ with an applied force $\mathbf{f}=f\hat{\mathbf{z}}$. Its energy contribution is $-fb\cos \theta$. Independent freely jointed orientations give the single-segment partition function and chain partition functionThe mean end-to-end projection is the force derivative of $k_{B}T\ln Z_N$:For $|u|\ll 1$, the Langevin function is $u/3+O(u^3)$. Isotropy therefore gives the vector responseThe covariance at zero force provides an independent route: $⟨R_i{R}_j{⟩}_0=N{b}^2{\delta }_{ij}/3$, and the Boltzmann weight's first-order term yields $⟨R_i⟩=f_j⟨R_i{R}_j{⟩}_0/(k_{B}T)$. This is a relation for the mean vector or its projection. The zero-force mean vector vanishes even though the chain has finite root mean square size $b\sqrt{N}$.

The controlled weak-force condition is $fb\ll k_{B}T$, not merely that a short part of a measured curve looks straight. With the supplied $b$, $k_{B}T/b\simeq 10.3\mathrm{pN}$ at $25^{\circ }\mathrm{C}$. A raw force near $30\mathrm{pN}$, as in the figure's labelled low-extension portion, would give $u\simeq 2.9$ and is not asymptotically weak for those literal segment parameters. A force offset or different effective model might change that interpretation, but no such correction should be invented from the plot.

On releasing the tensile load, the extended chain first retracts towards less aligned conformations because they have much greater configurational multiplicity. Viscous friction and internal relaxation determine the time course; stored free energy is dissipated as heat. An unconstrained ideal chain does not shrink to a point: its mean vector is zero but its root mean square end-to-end length is $b\sqrt{N}\simeq 2.37\mathrm{nm}$ for the supplied ideal-chain parameters.

A real polypeptide also has sequence-dependent interactions. If the original native structure is the stable state in the prevailing solvent and the molecule remains intact and capable of folding, it can subsequently refold, gaining favourable intramolecular and solvent free energy. The folded state is not selected by chain entropy alone, and formation of native contacts need not be instantaneous. Kinetic traps, proline isomerization, protein aggregation, misfolding, surface attachments or chemical damage may prevent complete recovery or produce hysteresis. Thus reversible refolding is a physically motivated possibility, not guaranteed simply by removal of force. This distinction connects the recoil of a stretched polymer to the thermodynamics discussed in Question 6(c).

The integrand is positive, so the minimum-variance importance distribution is its target-weighted probability density function, normalized by $\theta =\sqrt{e}$:By completing the square,Thus $q^{\star }$ is the multivariate normal distribution $N(-e_1,I_d)$, where $e_1=(1,0,\dots ,0)$ is the first standard basis vector. Its importance weight and weighted integrand areA variance cannot be negative, and this proposal makes every summand constant, proving optimality directly. It is admissible under the arbitrary-proposal condition even though its mean vector is not zero. Although optimal proposals usually involve an unknown normalizing integral, here both that integral and the resulting normal distribution are explicit.

A random vector $X=(X_1,\dots ,X_d)$ is a measurable map from one probability space to ${\mathbb{R}}^d$, equivalently a tuple of real random variables defined on the same space. Its joint probability distribution contains their dependence information. For example, if $B$ is a fair Bernoulli random variable, $(B,B)$ takes only $(0,0)$ and $(1,1)$, whereas two independent fair Bernoulli coordinates take all four pairs. Both have identical coordinate marginal distributions. A mean vector and covariance matrix summarize the first two moments when finite but generally do not determine the whole law.