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For events $A_1,\dots ,A_n$, the probabilistic inclusion-exclusion principle isTo prove it, fix an outcome lying in exactly $r$ events. Its total coefficient on the right isby the binomial theorem; outcomes in no event contribute zero. Taking expectations of this pointwise indicator identity proves the formula. Truncating after the pair terms gives the Bonferroni inequalities

For the final bound, let $X={\sum }_i{1}_{A_i}$ and $S=\mathbb{E}X={\sum }_i\mathbb{P}(A_i)$. The intersection assumption givesThe Cauchy-Schwarz inequality applied to $X{1}_{X>0}$ yields $S^2\le \mathbb{E}[X^2]\mathbb{P}(X>0)\le S+n-1$. Thereforeso the required universal constant may be taken as $\boxed{c=2}$. This is a second moment method estimate.

For a relativistic species the supplied relation $P=ϵ/3$ makes its entropy density proportional to $T^3$. The fermion-to-boson thermal integral ratio gives the factor $7/8$ for each fermionic degree of freedom, so the electromagnetic plasma immediately before electron-positron annihilation in cosmology hasAfter annihilation only the two photon polarizations remain, so $g_{\ast s,\mathrm{after}}=2$.

The neutrinos have already undergone neutrino decoupling. They therefore receive none of the electron-positron entropy and cool as $T_{\nu }\propto a^{-1}$. In the still-coupled electromagnetic plasma, cosmological entropy conservation gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$. If $T_d$ and $a_d$ denote the common temperature and scale factor just before annihilation, then at a later scale factor $a$,Dividing gives the Cosmic neutrino background temperature ratio

A $(1-\alpha )$ prediction interval is a pair of statistics $L(Y),U(Y)$ such that, for every $\beta$ and ${\sigma }^2>0$, an independent future response $Y^{\ast }$ at the specified input satisfies

In the normal linear model, assume $X$ has full column rank and writeFor a new observationindependent of the training sample, the prediction error isThe two terms on the right are independent centered normal random variables, whileConsequently, with the new-point regression leveragewe have

The assumption in the question givesThis variable is independent of $\hat{\beta }$; it is also independent of the new error ${\epsilon }^{\ast }$. Therefore it is independent of the standardized prediction error, and the defining normal-over-chi-square representation of Student's t-distribution givesIf $t_{n-p,1-\alpha /2}$ is the corresponding upper quantile, symmetry yieldsSolving these inequalities for $Y^{\ast }$ proves that the prediction interval in a normal linear model isIts center and endpoints depend only on the observed data and $x^{\ast }$, and the pivotal calculation proves the required coverage uniformly in the unknown $\beta$ and ${\sigma }^2$.