28K (Applied Probability)

For ordered times

0 < t_{1} < \dots < t_{n} < t

, the joint density together with

N_{t} = n

e^{- λ t} λ^{n}

. Dividing by

P (N_{t} = n) = e^{- λ t} (λ t)^{n} / n!

gives

n! / t^{n}

b

Conditionally on

T = t

N_{T}

is Poisson with mean

λ t

. Mixing over the exponential density gives

P (N_{T} = n) = \int_{0}^{\infty} e^{- λ t} \frac{( λ t ) ^{n}}{n !} ν e^{- ν t} d t = \frac{ν λ ^{n}}{( λ + ν ) ^{n + 1}} .

Thus it is geometric on ${0, 1, \dots}$ with success probability $ν / (λ + ν)$ .

An arrival at

s \leq t

remains active with probability

1 - (t - s)

when

0 \leq t - s \leq 1

, and zero later. By Poisson thinning, the detector count is Poisson with mean

m (t) = λ \int_{m a x (0, t - 1)}^{t} [1 - (t - s)] d s = {λ (t - t^{2} /2), λ /2, 0 \leq t \leq 1, t \geq 1.

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