(Apparently) Different Definitions of Statistical Independence I am reading John Mandel's book " The Statitical Analysis of Experimental Data).
In it, in Ch 5, p. 52, he describes Statistical Independence of two RVs $X,Y$ in terms of
independence in the respective errors of measurement.
He cites the example of a manufacture of spherical  balls from a homogeneous material such as steel, and balls being ball bearings of different sizes. We then weigh each specimen and measure its diameter using a micrometer.  the formular, :
$$ W =(\rho)(\pi/6)d^2 $$
Where $W$ is the weight of the ball,  $ \rho$ is the specific gravity, $d$ is the diameter of the ball,
Describes a _functional _ dependence between $W$ and $d$.
But he states that $W,d$ are not statistically dependent from each other, he says,
because errors $ \delta$ in measurement of $W$ do not have any effect in the error $ \epsilon$ of measurement of $d$, meaning that if we knew the exact value of $ \epsilon$, it would shed no value whatsoever on the value of $\delta$ and viceversa.
I am aware of the formal definitions of independence: RVs $X,Y$ are independent if for every pair of events $x$ in ( the range of ) $X, y$ in ( the range of ) $Y$: $$  P(X=x ,Y=y)= P(X=x)P(Y=y) $$ (1), i.e., the joint probability can be split into probabilities in terms of X,Y separately . Alternatively, $$P(X=x | y=Y)=P(X=x) ; P(Y=y| X=x)=P(Y=y)$$ (2) for all $x$ in the range of $X, y$ in the range of $Y$.
** Question is ** : Is Mandel's definition the same as the standard one? If so, how does it agree with (1), (2)?
 A: No, I do not think Mandel is using the usual definition of independence.
If $W$ and $d$ are random variables denoting the true, unknown values, and if $\hat{W}$ and $\hat{d}$ represent the measured values (according to what I'll assume is additive measurement error), then
$$
\hat{W} = W + \delta \quad \text{and} \quad \hat{d} = d + \epsilon .
$$
It's clear that the measurement errors $\delta$ and $\epsilon$ are assumed to be independent random variables, because the quoted passage says "if we knew the exact value of $\epsilon$, it would shed no value whatsoever on the value of $\delta$ and vice versa." This is exactly statement (2) of the definition of independence, as "knowing $\epsilon$" is the same as conditioning on $\epsilon$, and if this does not change the distribution of $\delta$ we can write
$$
p(\delta | \epsilon) = p(\delta) \quad \text{and} \quad p(\epsilon | \delta) = p(\epsilon).
$$
using $p(x)$ to denote the density function of a random variable $x$. In words, this means that the probability distribution of $\delta$ given the value of  $\epsilon$ is the same as the probability distribution of $\delta$ without knowing the value of $\epsilon$.
However, the same cannot be said for $W$ and $d$, because we know that the equation $W = Cd^2$ holds for a constant $C$. This means that learning the value of $d$ or of $W$ tells us exactly the value of the other variable. So if we learn the value of $d$, then $W$ is essentially no longer random. Formally, since $W$ is described as being a random variable, its probability density $p(W)$ is not a point mass. But the conditional density of $W$ given $d$ is a point mass, as
$$
p(W | d) = \Delta(W - Cd^2)
$$
where $\Delta$ denotes the Dirac delta function (terrible notation, I know, but little $\delta$ already denotes the measurement error of $W$). Since one is a point mass and the other is not, $p(W) \ne p(W | d)$ so we cannot say that $W$ and $d$ are independent according to the usual definition.
