Variable Rate Poisson Process Analysis

00:00 10 marks

Consider a non-homogeneous Poisson process where the rate function is given by \(\lambda(t) = 2 + \sin(t)\) for \(0 \le t \le \pi\). This process models the number of events occurring over time.

(a) Find the cumulative intensity function \(\Lambda(t) = \int_0^t \lambda(s)\,ds\) for \(t\) in \([0, \pi]\).

(b) Using your result from part (a), calculate the probability that exactly 3 events occur in the time interval \([0, \pi]\).

Part a

6 minutes 4 marks

Question

Evaluate the cumulative intensity function \(\Lambda(t) = \int_0^t (2 + \sin(s))\,ds\) for \(0 \le t \le \pi\).

Hint Hint

Hint

Integrate term by term, remembering the antiderivative of \(\sin(s)\) and applying the limits of integration.

Answer Answer
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Complete Answer

\(\Lambda(t) = 2t – \cos(t) + 1\)

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Part b

6 minutes 6 marks

Question

Using the result from part (a), calculate the probability that exactly 3 events occur in the interval \([0, \pi]\) for this non-homogeneous Poisson process.

Hint Hint

Hint

First, evaluate \(\Lambda(\pi)\) using the function found in part (a). Then, use the Poisson probability formula with \(k=3\) and \(\Lambda = \Lambda(\pi)\).

Answer Answer
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Complete Answer

\(P(N(\pi)=3)=\frac{e^{-(2\pi+2)}(2\pi+2)^3}{6}\)

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