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Unformatted text preview: ts into two at
rate , so q (i, i + 1) = i. To ﬁnd the transition probability of the Yule process
we will guess and verify that
pt (1, j ) = e t (1 ) for j tj 1 e i.e., a geometric distribution with success probability e
To explain the mean we note that
d
EX (t) = E X (t)
dt
j implies 1
t (4.13) and hence mean e t . E1 X (t) = e t . To check (4.13), we will use the forward equation (4.9) to conclude that if
1 then
p0 (1, j ) =
j pt (1, j ) + (j 1)pt (1, j 1)
(4.14)
t where pt (1, 0) = 0. The use of the forward equation here is dictated by the fact
that we are only writing down formulas for pt (i, j ) when i = 1. To check the
proposed formula for j = 1 we note that
p0 (1, 1) =
t e t = pt (1, 1) Things are not so simple for j > 1:
p0 (1, j ) =
t t e
+e t (1 (j ) e tj 1 1)(1 e ) tj 2 (e t ) 127 4.3. LIMITING BEHAVIOR
Recopying the ﬁrst term on the right and using e t = (1
the second, we can rewrite the righthand side of the above as
e t (1
+e ) t (j 1)(1 tj 2 (j t (1 e t )+ in 1) tj 1 e e e
)...
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 Spring '10
 DURRETT
 The Land

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