Resumen
For i.i.d. samples from a normal, uniform, or exponential distribution, we give exact formulas for the correlation between the sample variance and the sample range for all fixed n. These exact formulas are then used to obtain asymptotic expansions for the correlations. It is seen that the correlation converges to zero at the rate logn/n in the normal case, the 1/n rate in the uniform case, and the (logn)2/n rate in the exponential case. In two of the three cases, we obtain higher-order expansions for the correlation. We then obtain the joint asymptotic distribution of the interquartile range and the standard deviation for any distribution with a finite fourth moment. This is used to obtain the nonzero limits of the correlation between them for some important distributions as well as some potentially useful practical diagnostics based on the interquartile range and the standard deviation. It is seen that the correlation is higher for thin tailed and smaller for thick tailed distributions. We also show graphics for the Cauchy distribution. The graphics exhibit interesting phenomena. Other numerics illustrate the theoretical results.

Resumen
This article presents two expectation identities and a series of applications. One of the identities uses the heat equation, and we show that in some families of distributions the identity characterizes the normal distribution. We also show that it is essentially equivalent to Stein's identity. The applications we have presented are of a broad range. They include exact formulas and bounds for moments, an improvement and a reversal of Jensen's inequality, linking unbiased estimation to elliptic partial differential equations, applications to decision theory and Bayesian statistics, and an application to counting matchings in graph theory. Some examples are also given.