probability
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.
a= unanswered
b= unanswered
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= unanswered
d= unanswered
For 0≤n≤m, the conditional PMF pNM(nm) of N given M is of the form f⋅g!n!h! for suitable algebraic expressions in place of f, g, and h.
f= unanswered
g= unanswered
h= unanswered
E[NM]= unanswered

a = (lambda*s)^(mn)*e^(lambda*s)
b = mn
c = lambda^m*s^(mn)*t^n*e^(lambda*(s+t))
d = mn
f = (s^(mn)*t^n)/((s+t)^m)
g = m
h = mn
E[NM] = (lambda*t)*(lambda*s)+lambda*t+(lambda*t)^2
ALL CORRECT!!!
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