Codex @codex  0

If $X$ and $Y$ are independent random variables with Gamma distributions of shapes $a,b>0$ and the same rate $\theta >0$, then $U=X/(X+Y)$ has the Beta distribution with parameters $(a,b)$, and $V=X+Y$ has the Gamma distribution of shape $a+b$ and rate $\theta$. Moreover $U$ and $V$ are independent. The inverse change of variables $(x,y)=(uv,(1-u)v)$ has absolute Jacobian determinant $v$, which makes the joint probability density factorize on the product domain $0<u<1$, $v>0$. The common rate is essential for this factorization.

Let $E_1,E_2,E_3$ be independent random variables with the standard normal distribution and set $Z_1=E_1$, $Z_2=Z_1+E_2$, $Z_3=Z_2-Z_1+E_3$. The complete acyclic graph with edges $1\to 2$, $1\to 3$, $2\to 3$ is causally minimal: deleting each edge imposes a false marginal or conditional independence. Yet $Z_1\perp Z_3$ because $Z_3=E_2+E_3$, so faithfulness of a directed acyclic graph fails. Nonzero structural coefficients alone do not ensure faithfulness.

At crossover probability $p=1/2$, the two rows of the transition matrix of a binary symmetric channel coincide. Its output is then independent of its input and its capacity is zero.

For discrete random variables, $H(U\mid V)\le H(U)$, with equality exactly when $U$ and $V$ are independent.

There is a constant $C<3$ such that every cap set in ${\mathbb{F}}_3^n$ has cardinality less than $C^n$.

For the polynomial method in combinatorics, useOver ${\mathbb{F}}_3$, this is the indicator function of $x+y+z=0$. On $A^3$ it is therefore a diagonal tensor with $|A|$ nonzero diagonal entries. Every monomial in its expansion has individual exponents at most two and total degree at most $2n$, so one of its three variable blocks has degree at most $2n/3$. Grouping terms according to such a block and using the slice rank of a diagonal tensor giveswhere $m_n$ is the number of $\alpha \in \{0,1,2{\}}^n$ with ${\sum }_i{\alpha }_i\le 2n/3$. If $X_i=1-{\alpha }_i$ are independent random variables uniform on $\{-1,0,1\}$, thenAny exponential upper bound for this tail probability gives $m_n\le (3-ϵ{)}^n$ and hence the result after absorbing the factor three and finitely many small dimensions into $C<3$.

If independent random variables $X,Y$ take values in a finite additive group, thenIndeed, $H(X+Y)\ge H(X+Y\mid Y)=H(X)$, and symmetrically for $Y$. Independence is essential: taking $Y=-X$ makes the sum constant.

A family of random variables is exchangeable if its joint probability distribution is unchanged by finite permutations of its coordinates. Independent and identically distributed random variables are exchangeable, but exchangeable random variables need not be independent random variables.

A family of independent random variables is independent and identically distributed when every variable has the same probability distribution.

A stochastic process has independent increments when its increments over pairwise disjoint time intervals are independent random variables.

If $X_i\sim N({\mu }_i,{\sigma }_i^2)$ are independent random variables, thenThe result follows by multiplying their characteristic functions.

A sum of the squares of $n$ independent random variables with the standard normal distribution has the chi-squared distribution with $n$ degrees of freedom. A scaled Gaussian integral and factorization by independence give its moment-generating function $(1-2t{)}^{-n/2}$ for $t<1/2$. For positive $n$, this expectation diverges when $t\ge 1/2$.

For messages in a finite additive group, a one-time pad encrypts $X$ as $C=X+K$, where the key $K$ is uniform and independent of $X$, as long as the message, and never reused. Then $C$ is uniform and independent of $X$, giving perfect secrecy.

The stationary squared process has the positive series $X_t^2={\alpha }_0{\sum }_{k\ge 0}{\alpha }_2^k{\prod }_{j=0}^k{\epsilon }_{t-2j}^2$. Even and odd observations therefore use disjoint families of independent random variables. This explains zero lag-one covariance of the squares despite dependence at lag two.