Some Distributions Involving Bessel Functions

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Some Distributions Involving Bessel Functions
G M Kaufman
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1) of jj. Is to examine the expectation of y^ = e^ when z has the density (3. 3).
Use the asymptotic expansion (2. 10) for large values of z to write K, ([nC(z)/4] ) as proportional to \, [C(z)]"^e"^'"^^^^^' C([naz)M]*'^ (5. A) where ?(•) denotes the summand in C(z) of (2. 10). From (3. 3) the density of z times e^ may be written using (2. 10) as proportional to e ±00 J the convergence of (5. 5) depends on the exponent (isn + q)z - i5n[(z-m)^ + ^ . (5. 6) For any q > 0, (5. 6) approaches +« as
...z -*■ +". And so the fxmction (5. 2^ is divergent to -H» if q > 0. When q as z -> -«>.
The proof just given yields useful information about partial moments of iJ. : if q 2. ~"» (5*6) is bounded for all z -n. Consequently, in two action decision o — problems with acts whose expected values are linear in u.


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