![Create a graphical representation of a Poisson process, showing the number of arrivals over time intervals (0,t] and (0,t+s]. This should be a non-textual image, thus incorporating only visual aids such as dots or lines for indicating arrivals, an axis to represent time with highlighted intervals, and distinct color coding for rates λ, time intervals t and s. Do not include any text or algebraic expressions.](/images/questions/1052721/1024x1024.png)
Consider a Poisson process with rate λ. Let N be the number of arrivals in (0,t] and M be the number of arrivals in (0,t+s], where t>0,s≥0.
In each part below, your answers will be algebraic expressions in terms of λ,t,s,m and/or n. Enter 'lambda' for λ and use 'exp()' for exponentials. Do not use 'fac()' or '!' for factorials. Follow standard notation.
For 0≤n≤m, the conditional PMF pM∣N(m∣n) of M given N is of the form ab! for suitable algebraic expressions in place of a and b.
a=
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b=
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For 0≤n≤m, the joint PMF pN,M(n,m) of N and M is of the form cn!d! for suitable algebraic expressions in place of c and d.
c=
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d=
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For 0≤n≤m, the conditional PMF pN|M(n|m) of N given M is of the form f⋅g!n!h! for suitable algebraic expressions in place of f, g, and h.
f=
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g=
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h=
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E[NM]=
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a=(lambda*s)^(m-n)*e^(-lambda*s)
b= m-n
c= lambda^m*s^(m-n)*t^n*e^(-lambda*(s+t))
d= m-n
f= (s^(m-n)*t^n)/((s+t)^m)
g= m
h= m-n
E[NM]= (lambda*t)*(lambda*s)+lambda*t+(lambda*t)^2
a = λ^n * (λ*t)^m-n * exp(-λ*t) / (m-n)!
b = λ^m * s^n * exp(-λ*s) / (m-n)!
c = (λ*t)^n * exp(-λ*t) * (λ*t + λ*s)^m-n * exp(-λ*s) / (n!(m-n)!)
d = (λ*t + λ*s)^m * exp(-λ*t -λ*s) / (m-n)!
f = (λ*t)^n * exp(-λ*t) * (λ*t + λ*s)^m-n * exp(-λ*s) / (n!(m-n)!)
g = (m-n)!
h = (n!(m-n)!)
E[NM] = λ^2 * (t+s)
To find the expressions for a, b, c, d, f, g, h, and E[NM], we need to use the definitions of conditional and joint PMFs and the expected value.
a) The conditional PMF pM|N(m|n) of M given N is defined as the probability that there are m arrivals in (0,t+s] given that there are n arrivals in (0,t]. Since this is a Poisson process with rate λ, the number of arrivals in any interval follows a Poisson distribution.
The conditional PMF pM|N(m|n) can be expressed as:
pM|N(m|n) = P(M = m | N = n)
Using the properties of Poisson distribution, we know that the number of arrivals in disjoint intervals are independent. Therefore, the number of arrivals in (0,t+s] can be represented as the sum of the number of arrivals in (0,t] and the number of arrivals in (t,t+s].
Since we know that there are n arrivals in (0,t], the remaining s time interval will follow a Poisson distribution with rate λ*s. Therefore, the conditional PMF is given by:
pM|N(m|n) = P(M = m | N = n) = P(M-N = m-n | N = n) = P(Poisson(λ*s) = m-n) = (e^(-λs)*(λs)^(m-n)) / (m-n)!
Therefore, a = (e^(-λs)*(λs)^(m-n)) / (m-n)!
b) The conditional PMF pM|N(m|n) of M given N is also given by:
pM|N(m|n) = P(M = m | N = n) = P(N-M = n-m | M = m) = P(Poisson(λt) = n-m) = (e^(-λt)*(λt)^(n-m)) / (n-m)!
Therefore, b = (e^(-λt)*(λt)^(n-m)) / (n-m)!
c) The joint PMF pN,M(n,m) of N and M is defined as the probability that there are n arrivals in (0,t] and m arrivals in (0,t+s]. Since this is a Poisson process, the joint PMF can be calculated as the product of individual PMFs.
The joint PMF pN,M(n,m) can be expressed as:
pN,M(n,m) = P(N = n, M = m)
Using the properties of Poisson distribution, we can write:
pN,M(n,m) = P(N = n, M = m) = P(N = n) * P(M-N = m-n) = P(Poisson(λt) = n) * P(Poisson(λs) = m-n) = (e^(-λt)*(λt)^n) / n! * (e^(-λs)*(λs)^(m-n)) / (m-n)!
Therefore, c = (e^(-λt)*(λt)^n) / n!
And d = (e^(-λs)*(λs)^(m-n)) / (m-n)!
d) The conditional PMF pN|M(n|m) of N given M is defined as the probability that there are n arrivals in (0,t] given that there are m arrivals in (0,t+s].
Using the properties of conditional probability, we can write:
pN|M(n|m) = P(N = n | M = m) = P(N-M = n-m | M = m) = P(Poisson(λt) = n-m | Poisson(λs) = m) = [P(Poisson(λt) = n-m, Poisson(λs) = m)] / P(Poisson(λs) = m)
Using the properties of Poisson distribution, we can calculate the probabilities as:
P(Poisson(λt) = n-m, Poisson(λs) = m) = (e^(-λt)*(λt)^(n-m)) / (n-m)! * (e^(-λs)*(λs)^m) / m!
P(Poisson(λs) = m) = (e^(-λs)*(λs)^m) / m!
Therefore, f = (e^(-λt)*(λt)^(n-m)) / (n-m)!
And g = (e^(-λs)*(λs)^m) / m!
And h = m!
e) The expected value of NM can be calculated using the properties of a Poisson process. The expected value of a Poisson distribution with rate λ is equal to λ. Therefore, we can write:
E[NM] = E[N]*E[M | N] = (λt)*(λ(t+s)) = λ^2(t^2 + ts)
Therefore, E[NM] = λ^2(t^2 + ts)
To answer the questions about conditional PMF and joint PMF, let's first define some concepts related to Poisson processes:
- A Poisson process is a stochastic process that models the occurrence of events over time, where the events happen independently and at a constant average rate λ.
- The number of arrivals in a fixed time interval (0, t], denoted N, follows a Poisson distribution with mean λt.
- The number of arrivals in a longer time interval (0, t+s], denoted M, follows a Poisson distribution with mean λ(t+s).
Now let's start answering the questions:
a) The conditional PMF pM|N(m|n) represents the probability that there are m arrivals in the interval (0, t+s], given that there are n arrivals in the interval (0, t].
To compute this, we can use the following formula:
pM|N(m|n) = P(M = m | N = n) = P(N+m = n+m | N = n)
= P(N + m - n = m) = P(N2 = m)
Here, N2 represents a Poisson random variable with mean λs. So, a = λ^m * exp(-λs) / m!
b) Similarly, we can compute the conditional PMF pN|M(n|m), which represents the probability that there are n arrivals in the interval (0, t], given that there are m arrivals in the interval (0, t+s].
Using the same logic as before, we have:
pN|M(n|m) = P(N = n | M = m) = P(M+n = m+n | M = m)
= P(M + n - m = n) = P(M2 = n)
Here, M2 represents a Poisson random variable with mean λt. Thus, b = λ^n * exp(-λt) / n!
c) The joint PMF pN,M(n, m) represents the probability of having n arrivals in the interval (0, t] and m arrivals in the interval (0, t+s].
To compute this, we need to multiply the individual probabilities:
pN,M(n, m) = P(N = n, M = m) = P(N = n) * P(M = m)
Since N and M are independent, their respective probabilities are given by the Poisson distribution:
pN(n) = λ^n * exp(-λt) / n!
pM(m) = λ^m * exp(-λs) / m!
Thus, c = λ^(n+m) * exp(-λ(t+s)) / (n! * m!)
d) The expected value E[NM] represents the average number of arrivals in both intervals (0, t] and (0, t+s].
To compute this, we can use the fact that the expected value of the product of independent random variables is the product of their individual expected values:
E[NM] = E[N] * E[M]
Since N and M follow Poisson distributions, their expected values are:
E[N] = λt
E[M] = λ(t+s)
Therefore, E[NM] = (λt) * (λ(t+s)) = λ^2 * t * (t+s)
Hope this helps! Let me know if you have any further questions.