SEMAINE D’ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 135

tions of the notions of univariate distribution and bivariate relationship.

With reference to Figs. 2a and 3a, let x be the (unspecified)
molecular weight of sugar. Physical chemistry tells us that
the sugar molecules are crystals, all of which have the same
weight, w=3.01 x 107% grams, that is 180 times the atom
weight of hydrogen. If the weight pn could be measured exactly,
the situation would be as shown in Fig. 2a. In practice, the
weighting is subject to observational error, and if the errors
follow the normal distribution the measurements will be distributed
 as shown in Fig. 3a. The observed average x of this
distribution provides a point estimate of the unknown molecular
weight i. Next let x be the molecular weight of a polymere,
say a specific make of nylon. The nylon molecules are bands
of different length; that is, x is not a specific number, but a
variable subject to a specific distribution, say as shown in
Fig. 4a. Here jv denotes the mathematical expectation of the
distribution,

(37)

—
rt,

‘
‘

Distinguishing between the theoretical and the observed distribution,
 as illustrated in Fig. sa, the observed mean x gives a
point estimate of the theoretical mean p. Conceptually, the
dotted curve represents the distribution of a variable x =x* +¢
which is composed of a variable x* with the same distribution
as in Fig. 4a, and an observation error € which for fixed x* has
a distribution of the same type as in Fig. 3a. In the present
illustration it so happens that the molecular distribution can
only be observed indirectly, since the individual molecules are
too small for direct observation. Conceptually, we may think
of the observed distribution as referring to the individual molecular
 weights subject to observational error.
Comparing the situation in Figs. 2a and 3a with the more
general situation in Figs. 4a and 5a we note two simple instances
 of attenuated inference:

21 Wold - pag. 21