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Unformatted text preview: most (t + 1)t/2 + 1 of the binomial constraints are linearly
independent. Note that this means that the linear space of vectors satisfying these binomial
constraints has dimension at least 2t − (t + 1)t/2 − 1. This linear space has dimension 1
for t = 3 explaining the results in parts a) and b). It has a rapidly increasing dimensional
for t > 3, suggesting that the binomial constraints are relatively ineﬀectual for constraining
the joint PMF of a joint distribution. More work is required for the case of t > 3 because
of all the inequality constraints, but it turns out that this large dimensionality remains.
Exercise 2.6. Let h(x) be a positive function of a real variable that satisﬁes h(x + t) =
h(x) + h(t) and let h(1) = c. 84 CHAPTER 2. POISSON PROCESSES a) Show that for integer k > 0, h(k) = kc.
b) Show that for integer j > 0, h(1/j ) = c/j .
c) Show that for all integer k, j , h(k/j ) = ck/j .
d) The above parts show that h(x) is linear in positive rational numbers. For very picky
mathematicians, this does not guarantee that h(x) is linear in positive real numbers. Show
that if h(x) is also monotonic in x, then h(x) is linear in x > 0.
Exercise 2.7. Assume that a counting process {N (t); t≥0} has the independent and stationary increment properties and, for all t > 0, satisﬁes
n
o
e
Pr N (t, t + δ ) = 0
= 1 − ∏δ + o(δ )
n
o
e
Pr N (t, t + δ ) = 1
= ∏δ + o(δ )
n
o
e
Pr N (t, t + δ ) > 1
= o(δ ).
a) Let F0 (τ ) = Pr {N (τ ) = 0} and show that dF0 (τ )/dτ = −∏F0 (τ ). b) Show that X1 , the time of the ﬁrst arrival, is exponential with parameter ∏.
n
o
e
c) Let Fn (τ ) = Pr N (t, t + τ ) = 0  Sn−1 = t and show that dFn (τ )/dτ = −∏Fn (τ ). d) Argue that Xn is exponential with parameter ∏ and independent of earlier arrival times.
Exercise 2.8. Let t > 0 be an arbitrary time, let Z1 be the duration of the interval from
t until the next arrival after t. Let Zm , for each m > 1, be the interarrival time from the
epoch of the m − 1st arrival after t until the mth arrival.
a) Given that N (t) = n, explain why Zm = Xm+n for m > 1 and Z1 = Xn+1 − t + Sn .
b) Conditional on N (t) = n and Sn = τ , show that Z1 , Z2 , . . . are IID.
c) Show that Z1 , Z2 , . . . are IID.
Exercise 2.9. Consider a “shrinking Bernoulli” approximation Nδ (mδ ) = Y1 + · · · + Ym to
a Poisson process as described in Subsection 2.2.5.
a) Show that
µ∂
m
Pr {Nδ (mδ ) = n} =
(∏δ )n (1 − ∏δ )m−n .
n
b) Let t = mδ , and let t be ﬁxed for the remainder of the exercise. Explain why
µ ∂ µ ∂n µ
∂
m
∏t
∏t m−n
lim Pr {Nδ (t) = n} = lim
1−
.
m→1 n
δ →0
m
m 2.7. EXERCISES 85 where the limit on the left is taken over values of δ that divide t.
c) Derive the following two equalities:
µ∂
m1
1
lim
=
;
n
m→1 n m
n! and µ
∂
∏t m−n
lim 1 −
= e−∏t .
m→1
m d) Conclude from this that for every t and every n, limδ→0 Pr {Nδ (t)=n} = Pr {N (t)=n}
where {N (t); t ≥ 0} is a Poisson process of rate ∏.
Exercise 2.10. Let {N (t); t ≥ 0} be a Poisson process of rate ∏.
a) Find the joint probability mass function (PMF) of N (t), N (t + s) for s > 0.
b) Find E [N (t) · N (t + s)] for s > 0.
h
i
e
e
e
c) Find E N (t1 , t3 ) · N (t2 , t4 ) where N (t, τ ) is the number of arrivals in (t, τ ] and t1 <
t2 < t3 < t4 .
Exercise 2.11. An experiment is independently performed N times where N is a Poisson
rv of mean ∏. Let {a1 , a2 , . . . , aK } be the set of sample points of the experiment and let
pk , 1 ≤ k ≤ K , denote the probability of ak .
a) Let Ni denote the number of experiments performed for which the output is ai . Find
the PMF for Ni (1 ≤ i ≤ K ). (Hint: no calculation is necessary.)
b) Find the PMF for N1 + N2 .
c) Find the conditional PMF for N1 given that N = n.
d) Find the conditional PMF for N1 + N2 given that N = n.
e) Find the conditional PMF for N given that N1 = n1 .
Exercise 2.12. Starting from time 0, northbound buses arrive at 77 Mass. Avenue according to a Poisson process of rate ∏. Passengers arrive according to an independent Poisson
process of rate µ. When a bus arrives, all waiting customers instantly enter the bus and
subsequent customers wait for the next bus.
a) Find the PMF for the number of customers entering a bus (more speciﬁcally, for any
given m, ﬁnd the PMF for the number of customers entering the mth bus).
b) Find the PMF for the number of customers entering the mth bus given that the interarrival interval between bus m − 1 and bus m is x.
c) Given that a bus arrives at time 10:30 PM, ﬁnd the PMF for the number of customers
entering the next bus.
d) Given that a bus arrives at 10:30 PM and no bus arrives between 10:30 and 11, ﬁnd the
PMF for the number of customers on the next bus. 86 CHAPTER 2. POISSON PROCESSES e) Find the PMF for the number of customers waiting at some given time, say 2:30 PM
(assume that the processes started inﬁnitely far in the past). Hint: think of what happens
moving...
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 Spring '09
 R.Srikant

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