Well ! I can't let that one pass. Let it be noted that Robert Dawson does
not quote but paraphrases Humpty Dumpty. The correct quotation is:
===================
"When I use a word, " Humpty Dumpty said in rather a scornful tone, "it means
just what I choose it to mean--neither more nor less."
"The question is," said Alice, "whether you make words mean so many
different things."
"The question is," said Humpty Dumpty, "which is to be master--that's all."
===================
Also see J.B. Priestley, A Note on Humpty Dumpty, from "I for One", 1921;
and Roger Holmes, The Philosopher's Alice in Wonderland, Antioch Review,
1959.
As for the original problem, it suffices to observe that if U=12 (or 2) then
V must be zero.
At 9:54 AM -0400 2/7/00, Robert Dawson wrote:
>Guidi Chan wrote:
> > > A fair die is rolled 2 times. X1 and X2 is the # of points showing on
> > > 1st and 2nd rolls.
> > >
> > > U = X1 + X2; V = X1 - X2.
> > >
> > > Show that U and V are NOT independent.
>
>Howard Hoffman responded:
> > If you make a scatterplot of all possible values of U and V you will
> > discover that for every value of U the mean value of V is 0. In other
> > words, the slope of the regression of U on V is zero. This, for me is
>proof
> > that U and V are independent.
>
> "When I use a word" said Humpty Dumpty, "it means what I want it to
>mean.
>It all depends on who's going to be master, you or the word."
>
> The fact is that this is not the accepted meaning of independence. The
>accepted meaning of independence is that the conditional probability
>distribution of U does not depend on V. This can be rephrased usefully as:
>"there is no value of U that, if observed, would tell you anything about the
>value of V".
>
> As (this may have been a homework question but I presume it's past the
>due date by now, so I can be explicit) U=12 implies V=0 whereas U=11 makes
>this impossible, U and V are not independent. [End of proof.]
>
>
> Independence is actually quite hard to see from a scatterplot, as it
>is often hard to determine by eye if two conditional samples {U_i:V_i in
>(a1,a2)}
>and {U_i:V_i in (b1,b2)} have similar distributions or not when the numbers
>of
>data points differ significantly. For this purpose, I like to use an array
>of
>side-by-side boxplots of the dependent variable, one for each interval of
>the
>independent variable.
>
> Another frequently-confused-with-independence property of joint
>distributions is that the two marginal distributions are causally
>unrelated. This implies independence but does not follow from it (the
>canonical example involves the toss of two coins, with the events being
>A: heads on cent, B:exactly one head.)
>
> We have: causally independent => independent => regression slope = 0.
>
> -Robert Dawson
>
>
>
>
>
>
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===
Jan de Leeuw; Professor and Chair, UCLA Department of Statistics;
US mail: 8142 Math Sciences Bldg, Box 951554, Los Angeles, CA 90095-1554
phone (310)-825-9550; fax (310)-206-5658; email: [EMAIL PROTECTED]
http://www.stat.ucla.edu/~deleeuw and http://home1.gte.net/datamine/
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