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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 Gaussian scale mixture is a random variable $X=RZ$, where $Z$ has a standard normal distribution and is independent of a positive random scale $R$. Its probability density function is the average of the conditional normal densities. Random scales can produce larger kurtosis than a single normal law.

A real or complex measurable function is integrable for a measure $\mu$ when its absolute value has finite integral. For a random variable, this is integrability with respect to its probability measure, and makes its expectation finite.

A real or complex random variable $X$ is integrable when $\mathbb{E}|X|<\infty$. Its expected value is then finite, and conditional expectation preserves integrability.

The joint probability distribution of random variables $X_1,\dots ,X_n$ is the probability distribution of the random vector $(X_1,\dots ,X_n)$. It records both the marginal distributions and their dependence.

For a nonnegative random variable $X$, its Laplace transform is

For a random variable $Z$ and exponents $0<r<s$, the absolute moment norms satisfy $(\mathbb{E}|Z{|}^r{)}^{1/r}\le (\mathbb{E}|Z{|}^s{)}^{1/s}$, allowing infinity. For a finite $s$th moment, apply the Jensen inequality to the concave function $t^{r/s}$ and $|Z{|}^s$. The formula is for positive exponents; $r=0$ requires a separately defined limiting quantity.

For a nonnegative random variable with finite expected value and $\mathbb{P}(X>t)>0$, its mean residual life is the remaining conditional expected value after surviving past $t$. The tail integral formula for moments gives $m_X(t)={\int }_t^{\infty }\mathbb{P}(X>x)dx/\mathbb{P}(X>t)$. The exponential distribution has constant mean residual life, equal to its original mean. In excess of loss reinsurance, the total variance stationary condition for excess of loss sets a candidate retention equal to this quantity.

For a centered square-integrable martingale with $v=\mathbb{E}{M}_n^2$ and $\lambda >0$, apply the Doob maximal inequality for a nonnegative submartingale to $(M_k+c{)}^2$ for $c\ge 0$. Conditional Jensen inequality gives the submartingale property, and crossing $\lambda$ forces its square above $(\lambda +c{)}^2$. The bound $(v+c^2)/(\lambda +c{)}^2$ is minimized by $c=v/\lambda$. This extends the Cantelli inequality from one random variable to a martingale maximum, without requiring independent increments.