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Draw $P\sim \mathrm{Beta}(\alpha ,\beta )$, $X\mid P\sim \mathrm{Binomial}(n,P)$ and $Q\mid X\sim \mathrm{Beta}(\alpha +X,\beta +n-X)$, with $\alpha ,\beta >0$ and integer $n\ge 0$. The joint density of $(P,X,Q)$ is symmetric in $P,Q$, making them exchangeable random variables with identical Beta distribution marginal distributions. The posterior beta normalization constant depends on $X$ and must be retained. The correlation coefficient is $n/(n+\alpha +\beta )$: this constructs tunable dependence without altering either marginal.

For $X\sim N(\mu 1,C)$ with a known positive-definite matrix $C$, the maximum-likelihood estimator of $\mu$ is $\hat{\mu }=(1^T{C}^{-1}X)/(1^T{C}^{-1}1)$. It is unbiased with variance $(1^T{C}^{-1}1{)}^{-1}$, attaining the Cramer-Rao bound. It is also the best linear unbiased estimator. Pairwise admissible correlation coefficients alone do not ensure that $C$ is a valid covariance matrix; the entire matrix must be symmetric and a positive semidefinite matrix. Inverse-based formulas require positive definiteness.

For centered, unit-variance jointly Gaussian random variables with correlation coefficient $\rho$, the displayed identity holds. Realize the pair as projections onto unit vectors of angle $\theta =\arccos \rho$. Rotational symmetry of the two-dimensional Gaussian random vector direction makes sign disagreement have probability $\theta /\pi$. The expectation is therefore $1-2\theta /\pi =(2/\pi )\arcsin \rho$. Perfectly correlated and anticorrelated endpoints follow directly; the sign at zero does not affect the result.

For a bipartite elliptope matrix $X$, put $t=\log (1+\sqrt{2})$ and apply $\sinh (tx)$ within the two diagonal blocks and $\sin (tx)$ across them. Matching absolute power series coefficients give a positive semidefinite matrix by coefficient-dominated entrywise positivity, while $\sinh t=1$ gives unit diagonal. Gaussian hyperplane rounding then turns each cross-block correlation coefficient into $(2/\pi )\arcsin (\sin (t{X}_{ij}))=(2t/\pi )X_{ij}$ because $t<\pi /2$. The zero diagonal blocks of the bipartite objective eliminate every other contribution.

The principle terminal correlation does not identify adapted stock volatility exposes a false printed assertion for the correlation coefficient defined in the introduction. Take $r=0$, $S_0=S_0^{\prime }=1$, andThen $d{S}_t=S_{t}d{B}_t$ andHence ${\sigma }_t=1$ and ${\sigma }_t^{\prime }=S_t/(1+S_t)$ are bounded. Both terminal prices have positive finite variance and are perfectly correlated, because one is a positive affine function of the other. Nevertheless $\boxed{{\int }_0^1({\sigma }_t-{\sigma }_t^{\prime }{)}^{2}dt={\int }_0^1\frac{dt}{(1+S_t{)}^2}>0}$ on every continuous finite stock path. Equal initial prices fix the means, not the affine slope.

Here is the exact general conclusion and the intended qualified proof. Bounded volatilities make these zero-rate stochastic exponentials true square-integrable martingales. Writing $s_0=S_0=S_0^{\prime }$, perfect terminal correlation gives $S_1-s_0=c(S_1^{\prime }-s_0)$ for some $c>0$. Conditional expectation therefore gives $S_u-s_0=c(S_u^{\prime }-s_0)$ for every $0\le u\le 1$. Comparing stochastic integrals, the Itô isometry yieldsThis relation need not imply equality of the volatilities.

The terminal proportionality identifies bounded stock volatility criterion repairs the statement. If one additionally assumes equal terminal variances, then $c=1$ and the whole stock paths coincide, so positivity gives $\sigma ={\sigma }^{\prime }$ almost everywhere and the requested integral is zero. The same repair follows if terminal proportionality $S_1=c{S}_1^{\prime }$ is assumed: equal martingale means force $c=1$, after which the conditional-expectation and Itô isometry argument applies. This proves the intended conclusion under a sufficient additional hypothesis without treating centered correlation as uncentered proportionality.

Assume positive deterministic initial stock prices and finite, positive terminal variances, as required to define the correlation coefficient. The explicit stochastic-exponential solutions with constant volatilities giveThe common, possibly random, accumulated interest rate cancels. If $\sigma ={\sigma }^{\prime }$, this ratio is a positive constant, so the prices are perfectly correlated.

Conversely, perfect correlation would give $S_t=c{S}_t^{\prime }+d$ almost surely for constants $c>0,d$. If $\sigma \ne {\sigma }^{\prime }$, the displayed ratio is a nonconstant variable with a lognormal distribution with full support $(0,\infty )$. If $d>0$, the ratio would always exceed $c$, and if $d<0$ it would always be below $c$, both impossible. If $d=0$, the ratio would be constant, also impossible. Therefore $\boxed{\rho (S_t,S_t^{\prime })=1⟺\sigma ={\sigma }^{\prime }}$, whenever the stated terminal variances exist and are positive. No deterministic-interest assumption is needed. Degenerate terminal prices, for example zero volatilities and deterministic rates, have undefined correlation and must be excluded.

For centered square-integrable variables with positive variances, the correlation coefficient isPut $v=\sqrt{\mathbb{E}[X^2]}$ and $v^{\prime }=\sqrt{\mathbb{E}[(X^{\prime }{)}^2]}$. ThenThus $\boxed{\rho =1⟺X=(v/v^{\prime })X^{\prime }\text{ almost surely}}$, with positive proportionality factor. Conversely positive proportionality immediately gives correlation one. For variables with nonzero means, this criterion applies to their centered versions and yields an affine, not necessarily proportional, relationship. That distinction is essential for the positive stock prices below.

The law of iterated expectation yieldsOne may obtain $\mathrm{Var}(p_T)=v$ either from its unchanged marginal distribution proved below or directly from the law of total variance: the variance of its conditional mean is $nv/(s+n)$ and its expected conditional variance is $sv/(s+n)$. Hence the correlation coefficient is exactlyAlso $\mathbb{E}[(p_T-p_M{)}^2]=2vs/(s+n)\to 0$. The two probabilities become close while keeping their original beta marginals.

For a centered weakly stationary process with $\gamma (0)>0$, the autocorrelation function isFor $h\ge 2$, let $P$ be orthogonal projection in L2 space onto the linear span of $X_{t-1},\dots ,X_{t-h+1}$. The lag-$h$ partial autocorrelation function is the correlation coefficient of $X_t-P{X}_t$ and $X_{t-h}-P{X}_{t-h}$. It removes the linear contribution of the intervening observations. At lag one it is just $\rho (1)$. For nonsingular prediction covariance matrices, it is equivalently the last coefficient $a_{hh}$ in the order-$h$ linear predictor of $X_t$ from $X_{t-1},\dots ,X_{t-h}$. Equal residual variances, by stationarity, identify that coefficient with the residual correlation. If a residual has zero variance, this correlation is undefined; the nondegeneracy condition matters.

The three independent sample means satisfy ${\overline{Y}}_i\sim \mathcal{N}({\mu }_i,v_i)$. A linear transformation of their joint multivariate normal distribution is again normal. The two treatment contrasts have covarianceWriting $m_i={\delta }_i/s_i$ and $\rho =v_0/(s_1{s}_2)$, their joint distribution isThe positive correlation coefficient arises from the shared control arm. Under positive variances, $0<\rho <1$.

Couple the Wald test statistics as $W_i=Z_i+{\delta }_i/s_i$, where the joint centered bivariate normal distribution of $(Z_1,Z_2)$ is fixed. Increasing either ${\delta }_i$ enlarges, pointwise in this coupling, the event $\{W_1>c\text{ or }{W}_2>c\}$. Its probability is consequently nondecreasing in each treatment effect.

If both null hypotheses hold, then ${\delta }_1,{\delta }_2\le 0$, so this probability is maximal at the joint boundary $(0,0)$. To check the familywise error rate across all configurations, suppose only hypothesis $i$ is true. A Type I error then occurs only if $W_i>c$, with probability at most $1-\Phi (c)$, where $\Phi$ is the standard normal distribution function. This is no larger than the any-rejection probability at $(0,0)$, since that joint event contains $\{Z_i>c\}$. If neither null hypothesis is true, the familywise error rate is zero. Hence the least-favourable null configuration for Type I error iswhere ${\Phi }_2$ is the centered bivariate normal distribution function with correlation coefficient $\rho$.

For a causal autoregressive model of order $p$, the population partial autocorrelation function cuts off after lag $p$, whereas the autocorrelation function generally decays. For an invertible moving-average model of order $q$, the autocorrelation function cuts off after lag $q$, whereas the partial correlation coefficients generally decay. For a mixed autoregressive moving-average model, both generally decay. Exponential decay may alternate in sign or show damped oscillations. Sample correlation coefficients only approximate these patterns, so isolated crossings of the significance bands are not exact order tests.

In PDF Figure 1, the autocorrelation function alternates sign, with a large negative lag-one correlation coefficient and geometrically decreasing magnitude. The partial autocorrelation function has essentially one substantial spike, negative at lag one and about $-0.8$; later values mostly lie within the displayed bands. Thus a stationary AR(1) with a negative coefficient is the natural first model, with $\hat{\varphi }$ initially near $-0.8$. The trace fluctuates around an approximately constant mean and shows no evident deterministic trend.

Fit that model, compare nearby low-order alternatives using likelihood function and an Akaike information criterion or Bayesian information criterion, and inspect the residual autocorrelation function and residual variance. A few small later PACF spikes are expected in a plot with many lags. The roughly $\pm 1.96/\sqrt{T}$ white noise bands are a guide, rather than simultaneous guarantees for every lag.

The sign alternation also suggests concentration of spectral power near the high-frequency end: the AR(1) denominator $1+{\varphi }^2-2\varphi \cos \lambda$ is smallest near $\lambda =\pi$ when $\varphi <0$. From estimated $\gamma (0)$ and lag-one correlation coefficient one can estimate innovation variance as ${\hat{\sigma }}_{\epsilon }^2\approx \hat{\gamma }(0)(1-{\hat{\varphi }}^2)$. The persistence parameter controls the decay rate and long-run variance of a stationary process. These plots do not establish a normal distribution, independence, or the absence of nonlinear dependence; correlation coefficients of squared residuals can provide a separate variance diagnostic.