GATE CSE 1989 | Question: 4-viii

Question

$P_{n} (t)$ is the probability of $n$ events occurring during a time interval $t$. How will you express $P_{0} (t + h)$ in terms of $P_{0} (h)$, if $P_{0} (t)$ has stationary independent increments? (Note: $P_{t} (t)$is the probability density function).

Comments

Can you explain your solution? @goku4199 

---

 

 

Sudo_404_Div  In Sheldon Ross ,Poisson Distribution and Poisson Process are two different things.

The Poisson Distribution (Discrete): This counts the number of events happening in a fixed time/space. Example: "Number of customers arriving in 1 hour... This is Mass not density .

The Poisson Process (Continuous Time): This is a stochastic process where events occur continuously and independently at a constant average rate λ.

Note : The Poisson distribution (the count) strictly has a PMF. The density belongs to the Exponential or Gamma waiting times derived from the Poisson process.

 

 

 

---

For those why Abhrajyoti00 wrote λ= np

Answer 1

$P_0(t)$ denote the probability that no events happened in an interval of length $t.$$$P_0(t + h) = P_0(t) P_0(h)$$ This is because if there are no events in interval $[0,t+h]$ then there are no events in intervals 

1. $[0,t]$
2. $[t, t+h]$

These two intervals are non overlapping and it is given in question that $P_0(t)$ has stationary independent increments and so their probabilities are independent.

PS: One of the axioms of Poisson distribution is that the numbers of events in two nonoverlapping regions are independent.

Comments

What does stationary independent increment mean here?

---

http://www.randomservices.org/random/processes/Increments.html

---

Ans

---

Updated link : https://web.archive.org/web/20220611182400/http://www.randomservices.org/random/processes/Increments.html

Answer 2

Ans

Answer 3