probability

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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 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=- unanswered
g=- unanswered
h=- unanswered
E[NM]=- unanswered

  • probability -

    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

    ALL CORRECT!!!

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