Students arrive for the much dreaded JEE Advanced according to a Poisson process with rate lambda. For sanitization process they must stand in a queue,each student can take different time or same time compared with some other student for sanitization. Let us denote the time taken by i-th student as ${X}_{i}$.
${X}_{i}$ are independent identically distributed random variables. We assume that ${X}_{i}$ takes integer values in range 1,2, ... , n, with probabilities ${p}_{1},{p}_{2},\dots ,{p}_{n}$. Find the PMF for ${N}_{t}$, the number of students in sanitization queue at time t.
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Dawn Neal
Step 1
Given: Students arrive for the much dreaded JEE advanced according to Poisson process with rate Lamda $\left(\lambda \right)$ .
Let ${X}_{1},{X}_{2},\dots \dots$ Xnbe the time taken by students.${X}_{i}$ are independent and identically distributed and it takes integer values in range 1,2,3....n with probability ${p}_{1},{p}_{2},{p}_{3}\dots .pn$
The Poisson process is used in happening of repeatedly and independently in several settings
Step 2
Let ${N}_{t}$ be the number of students in sanitization queue at time [0,t].${N}_{t}$ is increasing integer valued, continuous time random process.
suppose the time [0,t]is divided into subinterval of width $\mathrm{△}t=\frac{t}{n}$
Step 3
Two assumptions
1) Probability of happing two events in subinterval is negligible, that is probability of happing of event in subinterval is either success or failure it gives only two outcomes hence it is Bernoulli trial.
2) Two events can not happen simultaneously either it give success or failure hence subinterval is independent of outcome in each interval, hence Bernoulli trials are independent.
The above two assumption shows that the counting process Nt can be binomial process (by result : The sum of independent Bernoulli trial approximated to Binomial distribution)
If the probability of number of students sanitize in queue in each subinterval is fixed P. then expected value of event happen in time [0,t] is given by binomial distribution is np. but event occurred at rate Lamda.
$\left(\lambda t\right)=\left(np\right)$
Let is fixed ,the binomial approaches to Poission distribution and its PMF is given by $N\left(t{N}_{t}\right)$
$P\left[N\left(t\right)=k\right]=\frac{{e}^{\lambda t}{\left(\lambda t\right)}^{k}}{k!}k=0,1,\dots$
where $e=2.71828$
$\lambda t=$ mean of distribution and $k=$ number of arrival in each subinterval
Answer: The PMF is given by,
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Neil Dismukes
Step 1
Solution:
Introduction to Poisson distribution:
A random variable X is said to have a Poisson distribution with a parameter $\lambda$ if its probability mass function (p.m.f) is as given below:
$P\left(x,\lambda \right)=\left\{\begin{array}{ccccc}\frac{{e}^{-\lambda }{\lambda }^{x}}{x!}& & x=012\dots \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\lambda >0& & \\ 0& & otherwise.& & \end{array}$
Here, students arrive for the much dreaded JEE advanced according to a Poisson process with the rate $\lambda$.
Denote the time taken by the student i as ${X}_{i}$.
Here, ${X}_{i}^{\prime }s$ are independent and identically distributed with probabilities ${p}_{1},{p}_{2},\cdots ,pn$.
Step 2
Find the PMF of ${N}_{t}$:
Here, Nt is the number of students in the sanitization queue at time t.
If $\lambda >0$ is fixed. The counting process $\left\{N\left(t\right),t\in \left[0,\mathrm{\infty }\right)\right\}$ is called a Poisson process with rate $\lambda$ under the below given conditions:
1. $N\left(0\right)=0$;
2. N(t) has independent increments;
3. The number of arrivals in any interval of length $\tau >0$ has Poisson $\left(\lambda \tau \right)$ distribution.
The PMF of Nt is as given below:
$N\left(t\right)=$ Number of students in the sanitization queue at time t
$N\left(t\right)\sim B\left(n,p\right)$
$n=\frac{t}{\delta }$
$p={p}_{1}+{p}_{2}+\cdots {p}_{n}$
Here, $p=\lambda \delta$
$np=\frac{t}{\delta }×\lambda \delta$
$=\lambda$
The PMF of N(t) converges to a Poisson distribution with parameter $\lambda$
$P\left(N\left(t\right)\right)=\left\{\begin{array}{ccccc}\frac{{e}^{-\lambda t}{\left(\lambda t\right)}^{x}}{x!}& & x>0& & \\ 0& & otherwise.& & \end{array}$