Class 21
Let \(\{N(t), t \ge 0\}\) be a Poisson process with rate \(\lambda\).
\(T_1 = {}\) the time of the first event.
\(T_2 = {}\) the time between the first and second event.
\(T_3 = {}\) the time between the second and third event.
\(\vdots\)
\(T_n = {}\) the time between the \((n-1)\)-st and \(n\)-th event.
\(\vdots\)
Then \(T_t \stackrel{\text{iid}}{\sim} \operatorname{Exp}(\lambda)\)
Suppose \(T_i \stackrel{\text{iid}}{\sim} \operatorname{Exp}(\lambda)\) for \(i = 1, 2, \ldots\).
Define \(\displaystyle S_n = T_1 + T_2 + T_3 + \cdots + T_n = \sum_{i=1}^n T_i\) for \(n = 1, 2, \ldots\)
Define \(S_0 = 0\), and let and \(N(t) = \max\{n; S_n \le t\}\) for \(t \ge 0\).
Then \(\{N(t), t\ge 0\}\) is a Poisson process with rate \(\lambda\).
Let \(\{N(t), t \ge 0\}\) be a Poisson process with rate \(\lambda\).
Suppose each of the arriving events satisfy a condition \(C\) with a probability \(p\), independently from all the other events.
Define \(N_C(t) = {}\) the number of events satisfying the condition \(C\) that arrived by time \(t\).
Then \(\{N_C(t), t \ge 0\}\) is a Poisson process with rate \(p\lambda\).
Customers arriving to a bank can be modeled as a Poisson process with rate \(\lambda = 15\) customers per hour. It is known that on average, two out of every 10 arriving customers come to open a new account.
What is the probability that between 1 pm and 6 pm, there will be exactly 18 customers who want to open a new account?
A probe is transmitting a constant stream of numbers collected from its sensors. Transmission errors can be modeled as a Poisson process with rate \(\lambda = 15\) errors per hour. Each error can be either:
The errors are independent of each other.
What is the probability that between 1 am and 9 am, there will be exactly 20 missing digits? Exactly 20 extra digits? Exactly 20 incorrect digits?
Let \(\{N_i(t), t\ge 0\}\) for \(i = 1, 2, \ldots, n\) be \(n\) independent Poisson processes with rates \(\lambda_1, \lambda_2, \ldots, \lambda_n\).
Define \(\displaystyle N(t) = \sum_{k = 1}^n N_k(t)\).
Then \(\{N(t), t \ge 0\}\) is a Poisson process with rate \(\lambda_1 + \lambda_2 + \cdots + \lambda_n\).
The ACME RoboCab company has a large fleet of self-driving vehicles. The vehicles are often involved in accidents. The company found out that the number of accidents with pedestrian involvement can be modeled as a Poisson process with rate 13.2 accidents per months, the number of accidents involving another vehicle can be modeled as a Poisson process with rate 5.7 accidents per month, and the number of accidents involving a stationary object can be modeled as a Poisson process with rate 2.1 accidents per month. It seems that the three types of accidents happen independently of each other.
What is the probability that there will be exactly 10 accidents during the first half of November?
Suppose that \(N(t) = 1\) for some time \(t\). In other words, \(T_1 \le t\). What is the conditional distribution of \(T_1\) in the interval \((0,t]\)?
Suppose that \(N(t) = n\) for some time \(t\). In other words, \(S_n \le t\). What is the (joint) conditional distribution of the arrival times \(S_1, S_2, \ldots, S_n\)?
Their joint conditional density function is
\[f\left(s_1, s_2, s_3, ..., s_n \mid N(t) = n\right) = \begin{cases} \frac{n!}{t^n} & \text{ if } 0 < s_1 < s_2 < \cdots < s_n < t\\ 0 & \text{ otherwise} \end{cases} \]