posted by JuanPro
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.
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.
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.
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