Given a poisson process Z(t) with a given rate lamda, k and m nonnegative integers and t and c real and positive numbers, calculate the probability:
P(Z(t-c)=m | Z(t)=k)
thank you
This is a poisson process so we have the following properties:
$\displaystyle [A]~Z_0 = 0$
$\displaystyle [B]~Z_t \sim Po(\lambda t)$
$\displaystyle [C]~Z_c - Z_b \sim Po(\lambda (c-b)) \text{ and is independent of } Z_b $
Start from the usual formula for conditional probabilities
$\displaystyle P(Z_{t-c} = m | Z_t =k) = \frac{P\left(Z_{t-c} = m \cap Z_{t} = k \right)}{P(Z_t)=k} $
The denominator is easy using property [B].
The numerator can be re-written as follows:
$\displaystyle P\left(Z_{t-c} = m \cap Z_{t} = k \right) = P\left(Z_{t-c} = m \cap Z_{t}-Z_{t-c} = k -m \right)$
using property [C], the events $\displaystyle Z_{t-c} =m$ and $\displaystyle Z_t -Z_{t-c} =k-m$ are independent, and hence
$\displaystyle P\left(Z_{t-c} = m \capZ_{t}-Z_{t-c} = k -m \right) = P\left(Z_{t-c} = m \right) \times P \left(Z_{t}-Z_{t-c} = k -m \right) $
Both of those terms follow a poisson distribution. (unless k-m is negative, then the probability is zero).
So the final answer is:
$\displaystyle P(Z_{t-c} = m | Z_t =k) = \frac{P\left(Z_{t-c} = m \right) \times P \left(Z_{t}-Z_{t-c} = k -m \right)}{P(Z_t = k)}$
Where:
$\displaystyle Z_t \sim Po(\lambda t)$
$\displaystyle Z_t - Z_{t-c} \sim Po(\lambda c)$
$\displaystyle Z_{t-c} \sim Po(\lambda (t-c))$
you can subsitute the formulae for the PMFs to finish. Dont forget that the probability is zero if m > k