Problem 5. Arrivals during overlapping time intervals
Consider a Poisson process with rate lambda.
Let N be the number of arrivals in the interval from 0 to t.
Let M be the number of arrivals in the interval from 0 to (t+s).
t is greater than 0,
s is greater or equal to 0
1. For numbers n and m that satisfy the constraint that 0 is less than n which is less than m, the conditional PMF p_M|N(m|n) of M given N is of the form a/(b factorial) for suitable algebraic expressions in place of a and b.
a = ?
b = ?
2. E[N*M] = ?
a : (lambda*s)^(m-n)*e^(-lambda*s)
b : m-n
2 : (lambda*t)*(lambda*s)+lambda*t+(lambda*t)^2
To find the conditional PMF of M given N, we need to use the properties of the Poisson process.
1. Conditional PMF p_M|N(m|n):
The conditional PMF p_M|N(m|n) represents the probability of having m arrivals in the interval from 0 to (t+s), given that there were n arrivals in the interval from 0 to t.
In a Poisson process, the number of arrivals in a given interval follows a Poisson distribution with parameter lambda multiplied by the length of the interval.
The number of arrivals in the interval from 0 to t follows a Poisson distribution with parameter lambda*t.
The number of arrivals in the interval from t to (t+s) is independent of the number of arrivals in the interval from 0 to t, and also follows a Poisson distribution with parameter lambda*s.
Therefore, the conditional PMF p_M|N(m|n) can be written as:
p_M|N(m|n) = P(M = m | N = n) = (e^(-lambda*s) * (lambda*s)^m) / m!
where lambda is the rate parameter of the Poisson process.
So, for the given values of m and n, the values of a and b are as follows:
a = e^(-lambda*s) * (lambda*s)^m
b = m!
2. Expected value E[N*M]:
The expected value E[N*M] can be calculated by considering the properties of the Poisson process.
In a Poisson process, the expected number of arrivals in an interval of length t is equal to the product of the rate parameter lambda and the interval length t.
Therefore, E[N] = lambda*t.
Similarly, the expected number of arrivals in an interval of length s is lambda*s.
Since the number of arrivals in the interval from 0 to t is independent of the number of arrivals in the interval from t to (t+s), the expected value of their product is simply the product of their individual expected values:
E[N*M] = E[N] * E[M] = (lambda*t) * (lambda*s) = lambda^2 * t * s
So, the expected value E[N*M] is lambda^2 * t * s.
To solve this problem, we need to understand the properties of the Poisson process and conditional probability. Here's the step-by-step explanation for finding the answers:
1. Conditional PMF p_M|N(m|n) of M given N:
The Poisson process defines the probability of having a certain number of arrivals in a given time interval. It satisfies the property that the number of arrivals in disjoint time intervals are independent.
Let's start with the conditional PMF p_M|N(m|n), which represents the probability of having m arrivals in the interval (0, t+s), given that there are n arrivals in the interval (0, t).
Since the number of arrivals in disjoint time intervals are independent, the number of additional arrivals in the interval (t, t+s) follows a Poisson distribution with rate lambda and time duration s.
Therefore, p_M|N(m|n) can be represented as the probability of having m-n additional arrivals in the interval (t, t+s), which can be calculated using the Poisson distribution:
p_M|N(m|n) = (e^(-lambda*s) * (lambda*s)^(m-n)) / (m-n)!
In this case, a = e^(-lambda*s) * (lambda*s)^(m-n) and b = m-n.
2. Expected value E[N*M]:
To find E[N*M], we need to calculate the joint probability mass function (PMF) of N and M. Since N and M are independent, their joint PMF is simply the product of their individual PMFs.
Given that N has n arrivals in (0, t) and M has m arrivals in (0, t+s), their joint PMF can be calculated as:
p_N,M(n,m) = p_N(n) * p_M(m)
The PMF of a Poisson process with rate lambda is given by:
p_N(n) = (e^(-lambda*t) * (lambda*t)^n) / n!
p_M(m) can be calculated similarly using the Poisson distribution with rate lambda and time duration t+s.
Then, E[N*M] can be calculated by summing over all possible values of n and m:
E[N*M] = Σ Σ (n * m * p_N(n) * p_M(m))
The first summation is over all possible values of n from 0 to infinity, and the second summation is over all possible values of m from n+1 to infinity.
Note: In practice, the summation is often truncated at some large value to get a finite approximation of the expected value.
I hope this explanation helps you understand how to obtain the answers to the given problem.