Number Theory

Mathematical Modeling for the Life Sciences by Jacques Istas

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By Jacques Istas

Proposing a variety of mathematical types which are at the moment utilized in lifestyles sciences should be considered as a problem, and that's exactly the problem that this ebook takes up. after all this panoramic examine doesn't declare to supply an in depth and exhaustive view of the numerous interactions among mathematical versions and existence sciences. This textbook offers a basic evaluation of lifelike mathematical types in existence sciences, contemplating either deterministic and stochastic versions and protecting dynamical platforms, online game idea, stochastic approaches and statistical tools. every one mathematical version is defined and illustrated separately with a suitable organic instance. ultimately 3 appendices on traditional differential equations, evolution equations, and likelihood are additional to give the chance to learn this booklet independently of different literature.

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54 2 Models for the Claim Number Process RT (a) Show that the random variables Tnn+1 λ(s) ds are iid exponentially distributed. s. and P (N (Ti ) − N (Ti −) > 1 for some i) = 0 . (9) Consider a homogeneous Poisson process N with intensity λ > 0 and arrival times Ti . , T0 = 0, Wi = Ti − Ti−1 are iid Exp(λ) inter-arrival times. Calculate for 0 ≤ t1 < t2 , P (T1 ≤ t1 ) and P (T1 ≤ t1 , T2 ≤ t2 ) . 1 for N . Calculate for 0 ≤ t1 < t2 , P (N (t1 ) ≥ 1) and P (N (t1 ) ≥ 1 , N (t2 ) ≥ 2) . 27). (10) Consider a homogeneous Poisson process on [0, ∞) with arrival time sequence (Ti ) and set T0 = 0.

The left bottom graph shows a moving average estimate of the intensity function of the arrivals µ(Tn ). Although the function is close to 1 the estimates fluctuate wildly around 1. 2. 22, where a QQ-plot15 of these data against the standard exponential distribution is shown. The QQ-plot curves down at the right. This is a clear indication of a right tail of the underlying distribution which is heavier than the tail of the exponential distribution. These observations raise the question as to whether the Poisson process is a suitable model for the whole period of 11 years of claim arrivals.

The increments M ((x, x + h] × (t, t + s]) = #{i ≥ 1 : (Xi , Ti ) ∈ (x, x + h] × (t, t + s]} , • x, t ≥ 0 , h, s > 0 , are Pois(F (x, x + h] µ(t, t + s]) distributed. For disjoint intervals ∆i = (xi , xi + hi ] × (ti , ti + si ], i = 1, . . , n, the increments M (∆i ), i = 1, . . , n, are independent. From measure theory, we know that the quantities F (x, x + h] µ(t, t + s] determine the product measure γ = F × µ on the Borel σ-field of [0, ∞)2 , where F denotes the distribution function as well as the distribution of Xi and µ is the measure generated by the values µ(a, b], 0 ≤ a < b < ∞.

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