Math 171 Homework #4
3.3
a. Since Xand Yare Poisson, then Xti−Xsiand Yti−Ysiare independent
random variables in non-overlapping intervals ti−si. Thus Xti−Xsi+Yti−Ysi
are independent.
If events in Xand Yoccur one at a time, then X+Yoccurs one at a time.
The only other possibility is that an event in Xand Yoccur at the same
time. But this occurs with probability lim∆t→0λ1λ2(∆t)2.
Events in Zoccur with rate (λ1+λ2).
b. We get λ1
λ1+λ2.
c. The probability that someone has called phone 1 is 1 −e−λ1t. The prob-
ability that someone has called both phones is then (1 −e−λ1t)(1 −e−λ2t) =
{T≤t}.
3.5
A=
1 2
1−1 1
24−4
=1 1
1−40 0
0−54/5 1/5
1/5−1/5
Pt=etA
=1 1
1−40 0
0e−5t4/5 1/5
1/5−1/5
=4/5 1/5
4/5 1/5+e−5t1/5−1/5
−4/5 4/5
3.7
Consider two arbitrary states i,j. Because the system is irreducible, we know
that there exit z1, z2, . . . , xk∈Swith α(x, z1), α(z1, z2), . . . , α(zk−1, zk), α(zk, y)
all strictly positive. We can divide time into (k+ 1) subintervals and write
P{Xt=j|X0=i} ≥ P{Xt/(k+1) =z1|X0=i}P{X2t/(k+1) =z2|Xt/(k+1) =z1} · · ·
· · · P{Xkt/(k+1) =zk|X(k−1)t/(k+1) =zk−1}P{Xt=j|Xkt/(k+1) =zk}
≥α(i, z1)(t/(k+ 1))α(z1, z2)(t/(k+ 1)) · · · α(zk, j )(t/(k+ 1))
≥0
3.8
