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**Unformatted text preview: **roc univariate data=prob2_7 plots;
histogram fatalities;
run; 2.7c The problem asks us to verify the formula,
xj +1 = xj +
¯
¯ xj +1 − xj
¯
.
j+1 Note that this is a formula for xj +1 in terms of xj and xj +1 . To see why
¯
¯
the formula should be correct, let’s imagine that we had only xj and xj +1
¯
and that we wanted to compute xj +1 : how would we proceed?
¯
Well, to compute xj +1 , we would want to start with the sum
¯
x1 + x2 + . . . + xj + xj +1 .
And we can get the ﬁrst j terms in the sum by multiplying xj by j . That is,
¯
we can get the whole sum as
j xj + xj +1 ,
¯
so that we could get xj +1 as
¯
j xj + xj +1
¯
.
j+1
So the question comes down to whether
xj +1 − xj
¯
j xj + xj +1
¯
= xj +
¯
.
j+1
j+1
To see make the comparison easier, let’s write the right hand side as a
fraction with denominator j + 1, so that our question becomes whether
j xj + xj +1
¯
(j + 1)¯j + xj +1 − xj
x
¯
=
.
j+1
j+1
And a tiny bit of algebra in the numerator on the right hand s...

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