Question

Let {N (t)} be a Poisson process with rate λ and arrival times S1, S2, . . .. Suppose that, for

a fixed time t, we are given that N (t) = 2.
(a) Find the conditional pdf for the second arrival time given that N (t) = 2. (Hint: Start
with the conditional cdf.)

Answer 1

To find the conditional pdf for the second arrival time given that N(t) = 2, we need to start with the conditional cdf. Use the properties of the exponential distribution and Poisson distribution to find the conditional pdf of T2 given that N(t) = 2.

Step-by-step explanation:

To find the conditional pdf for the second arrival time given that N(t) = 2, we first need to start with the conditional cdf.

Let T1 represent the time of the first arrival and T2 represent the time of the second arrival. We want to find the probability density function (pdf) of T2 given that N(t) = 2.

We can start by finding the conditional cumulative distribution function (cdf) of T2 given that N(t) = 2. Since N(t) = 2, it means that the first two arrivals occur within the time interval t. So, the conditional cdf is P(T2 ≤ x | N(t) = 2) = P(T1 ≤ x, T2 ≤ x).

From here, you can continue with the calculations and use the properties of the exponential distribution and Poisson distribution to find the conditional pdf of T2 given that N(t) = 2.