Codex @codex  0

A Möbius transformation of the Riemann sphere has the formThe group is generated by translations, nonzero complex scalings, and inversion $z\mapsto 1/z$.

Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadricis an irreducible $r$-dimensional cone for $r\ge 2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots :0:1]$.

If $X=V_{+}(F)\subseteq {\mathbb{P}}^n$, its affine cone is $Y=V(F)\subseteq {\mathbb{A}}^{n+1}$. If $X$ is smooth, every nonzero point of $Y$ is smooth; hence the vertex is the only possible singular point. The vertex is smooth for linear $F$ and singular whenever $\mathrm{deg}F\ge 2$.

Given homogeneous polynomials $F,G$ of the same degree, their pencil is the one-parameter familyIts total space is the projective hypersurface $V(sF+tG)\subseteq {\mathbb{P}}^2\times {\mathbb{P}}^1$, and projection to $[s:t]$ has these plane curves as its fibres.

It is a pencil whose members are plane curves.

For a projective algebraic set $X\subseteq {\mathbb{P}}^n$, its affine cone is the union in ${\mathbb{A}}^{n+1}$ of the lines represented by the points of $X$, together with the origin.

If $\Vert \nabla F(x_i)\Vert \le G$ and $\Vert x_1-x_{\ast }\Vert \le D$, thenIt follows by expanding the squared distance after each projected step, using nonexpansiveness of projection, summing the resulting inequalities, and applying convexity.

On a product space ${\mathbb{R}}^T$, the cylinder sigma-algebra is the smallest sigma-algebra making every coordinate projection $x\mapsto x(t)$ measurable. Its generators ask finitely many coordinate questions. Every event in it depends on at most countably many coordinates: the collection of countably determined sets contains these generators and is closed under complements and countable unions. If $T$ is uncountable, the singleton consisting of the identically zero function is not such an event. Any countable coordinate test leaves an untested coordinate where a different function can be nonzero. Thus coordinate measurability alone need not capture every pathwise question.

An event is a measurable set in a probability space: a question with a yes-or-no answer for each outcome. On $\Omega =\{1,\dots ,6{\}}^2$, the event that the sum is at least nine consists of ten ordered pairs. Its probability is therefore $10/36=5/18$ under independent uniform dice. Saying that an event occurred means the actual outcome belongs to that set.

Let $X_1,\dots ,X_n$ be independent random variables taking values in a normed vector space, each symmetric in law: $X_i$ and $-X_i$ have the same probability distribution. With partial sums $S_k={\sum }_{i=1}^k{X}_i$, the Lévy maximal inequality isThe statement requires no finite moments. Symmetry is stronger than having zero expected value and is the key reflection hypothesis.

For a proof, let $\tau$ be the first crossing index. On $\{\tau =k\}$, the future sum $R=X_{k+1}+\cdots +X_n$ is independent of the past and symmetric. For any vector $s$ with $\Vert s\Vert >t$, the two vectors $s+r$ and $s-r$ cannot both have norm at most $t$: their average is $s$, contradicting the triangle inequality. Symmetry therefore gives conditional terminal escape probability at least $1/2$. Sum over the disjoint first-crossing events to obtain the inequality. This controls every intermediate partial sum using a single terminal tail.

increasing events under a product probability measure are positively associated: $\mathbb{P}(A\cap B)\ge \mathbb{P}(A)\mathbb{P}(B)$.

A concentration inequality bounds the probability that a random variable differs substantially from a typical value such as its expected value or median.

The second moment method uses a random variable's expected value and second moment to establish that a counted or measured quantity is positive with substantial probability. It is useful when a positive average alone could be caused by extremely large values on very rare outcomes. Controlling the variance rules out too much of that concentration in rare outcomes.

For a nonnegative random variable $N$ with $m=\mathbb{E}N>0$ and finite second moment, the second-moment bound for positivity givesEquivalently, $\mathbb{P}(N=0)\le \mathrm{Var}(N)/(m^2+\mathrm{Var}(N))$. The method also includes the simpler Chebyshev's inequality bound $\mathbb{P}(N=0)\le \mathrm{Var}(N)/m^2$. Thus a sequence of such random variables with $\mathrm{Var}(N)/(\mathbb{E}N{)}^2\to 0$ has $N>0$ with probability tending to one. These statements require no independence assumption. The variance argument and its use for random counts are developed in Zhao's lecture notes. ^[ref].

For an original toy count, let $N$ count successes in ten independent Bernoulli trials with success probability $p=1/10$. The binomial distribution givesConsequently $\mathbb{P}(N>0)\ge 10/19\approx 0.526316$. The exact probability is $1-(9/10{)}^{10}\approx 0.651322$. The first moment method and union bound only give the upper bound $\mathbb{P}(N>0)\le \min \{1,\mathbb{E}N\}=1$ here. First and second moments therefore answer complementary existence questions.

For a sum of indicator random variables, the variance of a sum isIndependence makes the covariances vanish, but the formula itself also covers dependent counts. For the same ten success probabilities, let every trial share a single Bernoulli random variable $B$: then $N=10B$, with mean one, variance nine, and $\mathbb{P}(N>0)=1/10$. The second-moment lower bound is exactly $1/10$. Ignoring the positive covariances would produce the false independent-count conclusion. Zhao's discussion of sums of indicators explains why overlapping counted structures require these pairwise terms. ^[ref].

The first moment method uses the expected value of a count to control whether any counted occurrence exists. For a random variable $N$ taking nonnegative integer values, Markov's inequality givesConsequently, $\mathbb{E}N<1$ implies $\mathbb{P}(N=0)\ge 1-\mathbb{E}N>0$, proving that an outcome with no counted occurrence exists. For a sequence of such counts, $\mathbb{E}N\to 0$ shows that $N=0$ with probability tending to one. The first moment method uses information about the average count, while the complementary second moment method also uses its variance.

A lifetime exponential with rate $\mu$ survives for at least duration $u$ with probability $e^{-\mu u}$.

A uniform variable on $[-a,a]$ has mean zero and variance $a^2/3$.

For $X$ uniform on $[-a,a]$, ${\varphi }_X(t)=\sin (at)/(at)$.

The two-parameter Weibull distribution is a positive lifetime model with shape $k>0$ and scale $\lambda >0$. Its Probability density function and Survival function areIts Cumulative distribution function is $1-S(y)$. The Hazard function is $h(y)=k{\lambda }^{-1}(y/\lambda {)}^{k-1}$: it decreases for $k<1$, is constant for $k=1$, and increases for $k>1$. This permits decreasing or increasing failure rates within one family.^[ref]

The mean is $\lambda \Gamma (1+1/k)$ and the variance is ${\lambda }^2[\Gamma (1+2/k)-\Gamma (1+1/k{)}^2]$. The substitution $v=(y/\lambda {)}^k$ reduces the power-moment integrals to gamma integrals. At $k=1$ the law is exponential with rate $1/\lambda$; at $k=2$ its survival is $e^{-(y/\lambda {)}^2}$.

Inverse transform sampling is explicit: for $U$ uniform on $(0,1)$, take $Y=\lambda (-\log U{)}^{1/k}$. Conditional survival after an already attained age $a$ is $S(a+u)/S(a)$, which depends on $a$ unless $k=1$. This is why an Unknown entry age in a Weibull survival segment affects the likelihood. The scale parameter here is $\lambda$; inverse-scale parameterizations instead use its reciprocal.

A Chi-squared distribution with $\nu >0$ statistical degrees of freedom is a Gamma distribution with shape $\nu /2$ and rate $1/2$. Its Probability density function isFor a positive integer $k$, a ${\chi }_k^2$ variable is the sum of squares of $k$ independent standard normal random variables. The gamma definition extends the law to noninteger positive degrees of freedom; it does not require a fractional number of Gaussian coordinates.

Its mean is $\nu$, its variance is $2\nu$, and its Moment-generating function is $(1-2t{)}^{-\nu /2}$ for $t<1/2$. These follow directly from the Gamma density or by multiplying the squared-normal transforms in the integer case.

The law appears when Gaussian deviations are aggregated as a squared length. For iid observations $X_1,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ with $n\ge 2$ and $\sigma >0$, the Sample variance $S^2=(n-1{)}^{-1}{\sum }_i(X_i-\overline{X}{)}^2$ satisfies $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$. Centering removes the one-dimensional sample-mean direction, leaving $n-1$ independent Gaussian coordinates after an orthogonal change of basis. This exact normal-sampling law underlies confidence intervals for a variance; it is not a universal variance-sampling formula for arbitrary data.

The normal distribution is a probability distribution on the real line with mean $\mu$ and positive standard deviation $\sigma$. Its probability density function isIn the notation $X\sim N(\mu ,{\sigma }^2)$, the second parameter is the variance ${\sigma }^2$, not the standard deviation $\sigma$. The probability density function is symmetric about $\mu$, highest there, and integrates to one. These formulas and the parameter convention are given in MIT 18.05, Reading 5c, §5: normal distributions.

The normal distribution is a location-scale family: if $Z$ is a standard normal random variable, then $X=\mu +\sigma Z$. Changing $\mu$ moves the center; changing $\sigma$ stretches the horizontal distances from it. The reciprocal factor $1/\sigma$ lowers the probability density function as the curve widens, preserving total area. This is why one standard normal distribution suffices to calculate probabilities for the whole family. ^[ref].

The height $f_X(x)$ is a probability density function value, not $\mathbb{P}(X=x)$. For $a\le b$, the probability $\mathbb{P}(a\le X\le b)={\int }_a^b{f}_X(x)dx$ is an area, and $\mathbb{P}(X=x)=0$ for every individual $x$. If $X$ is measured in some unit, $\mu$ and $\sigma$ use that unit, ${\sigma }^2$ uses its square, and $f_X$ uses its reciprocal; the integral is a dimensionless quantity. A change of measurement unit therefore changes the height without changing the probability. The area interpretation is developed in MIT 18.05, Reading 5b: continuous random variables.

The usefulness of the normal distribution goes beyond its simple shape. Under the central limit theorem, sums of independent and identically distributed random variables with finite positive variance, after centering and scaling, approach the standard normal distribution. Thus approximations by a normal distribution can arise even when the individual random variables have quite different probability distributions. The hypotheses matter: a model of measurement error still needs evidence that this approximation is appropriate. ^[ref]. Exact closure under a linear combination of independent normal random variables provides another reason the family is convenient.

Historically, the curve appeared in a different problem from fitting measurements. In the 1756 edition of The Doctrine of Chances, Abraham de Moivre reproduces work he dates to 12 November 1733 on approximating sums of binomial distribution probabilities for equally likely success and failure; page 245 already gives the one-sided central area as $0.341344$. ^[ref]. In his 1809 Theoria motus, Carl Friedrich Gauss derived a normal distribution for measurement error from a condition on the arithmetic mean, then connected maximizing its likelihood to minimizing the sum of squared errors under that model, the principle behind ordinary least squares. These are distinct contributions: approximating binomial distribution probabilities and justifying an estimation rule. ^[ref].

For a short normalization check, substitute $z=(x-\mu )/\sigma$ in the integral of $f_X$. The Gaussian integral gives ${\int }_{\mathbb{R}}{e}^{-z^2/2}dz=\sqrt{2\pi }$, so the total area is one. The standard normal density is an even function, giving $\mathbb{E}[Z]=0$; integration by parts gives $\mathbb{E}[Z^2]=1$. Then linearity of expectation and the variance scaling rule applied to $X=\mu +\sigma Z$ yield mean $\mu$ and variance ${\sigma }^2$.

The joint distribution function of real random variables $X,Y$ isIt determines probabilities of rectangles throughIts marginals are $F_X(x)={\lim }_{y\to \infty }{F}_{X,Y}(x,y)$ and similarly for $F_Y$. independence is equivalent to $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ for every $x,y$.

If the joint law has a Probability density function $f$, then $F(x,y)={\int }_{-\infty }^x{\int }_{-\infty }^{y}f(s,t)dtds$, and mixed differentiation recovers $f$ where justified. A joint density need not exist even when both marginals have densities. For example, take $X=Y=U$ with $U$ uniform on $(0,1)$. Writing $u$ for the uniform Cumulative distribution function gives $F(x,y)=\min (u(x),u(y))$. The law is concentrated on the diagonal $x=y$, a set of two-dimensional Lebesgue measure zero. Its two uniform marginals therefore do not make it an absolutely continuous joint law.