38 Hint: See Problem 33, The Gompertz growth equation normalized so hat proporti

Question

38 Hint: See Problem 33, The Gompertz growth equation normalized so hat proportion (ie, y = 1 is an equilibrium and upper bound) is given by the variable y has the interpretation of a pr dy--ry ln(y) dt This equation can be used to model a variety of pop- ulation processes, including tumor growth (y is pro- portion of maximum size), population growth (yis proportion of environmental carrying capacity), and acquisition of new technologies, as illustrated in the following example. The Gompertz equation has been used to model mobile phone uptake, where y(t) is the fraction of individuals who have a mobile phone by time t (say, in years) and r is a parameter that can be fitted to the actual data. Using this model, we can derive a prob- ability density function that represents the time at which an individual acquired her first mobile phone.

Explanation / Answer

Here dy/dt = -ry ln(y)

here given is

r = 1 and y(0) = 1/e

so dy/dt = -y ln(y)

dy/(ylny) = - dt

ln (ln l y l ) = -t + C

for t = 0 ; y = 1/e

ln (ln (l 1/e l) = C

C = 0

so y(t) = e-e^t

(b) F(t) = 1 - y(t)

here F(t) will be cdf as F( -) = 1 - y(- ) = 0

and F( ) = 1 - y( ) = 1

so this is a Cumulative Distribution function. where  - < t <  

(c) F(t) = 1 - e-e^t

f(r) = dF(t)/dt = e-e^t et

f(r) = e-e^t et

(d) here Pr( t = 2years) = e-e^t = e-e^2 * e2 = 0.004566

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