Showing papers in "Annals of Mathematical Statistics in 1936"

On the Frequency Function of $xy$

Abstract: Given the distribution function of x and y, what can be said of the distribution of the product xy? The author has had two inquiries during the last two years, one from an investigator in business statistics and the other from a psychologist, concerning the probable error of the product of two quantities, each of known probable error. There seems to be very little in the literature of mathematical statistics on this question. If x and y are independent and are each distributed according to the same normal frequency law, it is well known that the distribution function of

Sheppard's Corrections for a Discrete Variable

Abstract: In the Annals of Mathematical Statistics,' J. R. Abernethy gave a derivation of the corrections to eliminate the systematic errors in the moments of a discrete variable due to grouping. It is the purpose of this note to considerably shorten and simplify the derivation of these corrections by an adoption of a device used by R. A. Fisher (not published so far as I know) in the case of the ordinary Sheppard's corrections. Let us suppose that m consecutive values of the discrete variable in question are grouped in a frequency class of width k. The m smaller intervals of width k/m go to make up the class width k, the actual points representing the m values of the variable being plotted at the centers of the sub-intervals. Now let us suppose that each of m consecutive boundary points of the sub-intervals is as likely to be chosen as a boundary point of the larger intervals as any other. Then, if xi is the class mark of the i-th frequency class, for any true value, x, of the discrete variable included in this frequency class, we hav e