# Learning and Memory: The Behavioral and Biological Substrates

By Isidore Gormezano; Edward A. Wasserman | Go to book overview

APPENDIX A : TAU-GAMMA HYPOTHESIS

We require the distribution of

rn = τ + tn + 1-tn, (A1)

where the ti are 2-fold general gamma variates, t = a + b, with

f(a) = α and f(b) = βe-βb, α ≠ β (A2)

McGill and Gibbon ( 1965) have analyzed this convolution in some detail. For two components, it has density

g2(t) = c1αe-αt + c2βee-βtt (A3)

where

c1 = 1/(1-α/β), c2 + 1/(1-β/α). (A4)

To find the ti difference distribution in (A1), let x = tn + 1-tn and consider the event x>T, T>0. We fix tn, and

1-Fx(T) ≡ P(x > T) = P(tn + 1> T + tn) ≡ 1 - G2 (T + tn),

known to be given by

1 - G2(y) = c1e-αy + c2e-βy.

Setting y = T+tn and integrating,

which gives

1 - Fx(T)=c12e-αT[(1/2)-α/(α+β)] + c22e-βT[(1/2)-β/(α+β)].

Some algebraic simplification shows

1 - Fx(T) = (1/2)(c1d1e-α|T| + c2d2e-β|T|), (A5)

where the ci are as above (A4), and

d1 = 1/(1 + α/β), d2 = 1/(1 + β/α). (A6)

But a strictly symmetrical argument for x

Differentiating A5, x is seen to have density

f(x) = (1/2)(c1d1αee-αU+007Cx| + c2d2βe-β|x|), (A7)

with, of course, Fr(T) = Fx(T-τ) and fr(r) = fx(r-τ).

-128-

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