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ST3236: Stochastic Process Tutorial 10

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ST3236: Stochastic Process Tutorial 10 TA: Mar Choong Hock Email: g0301492_at_nus.edu.sg Exercises: 11 Question 1 Question 1 Question 2 Question 2 Question 2 Question 2 ... – PowerPoint PPT presentation

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Title: ST3236: Stochastic Process Tutorial 10


1
ST3236 Stochastic ProcessTutorial 10
  • TA Mar Choong Hock
  • Email g0301492_at_nus.edu.sg
  • Exercises 11

2
Question 1
For i 1, , n. Let Xi(t) t gt 0 be
independent Poisson processes, each with the
same parameter ?. Find the distribution of the
first time that at least one event has occurred
in every process.
3
Question 1
Note You can give either the c.d.f. or the
p.d.f. as the answer for this case, the c.d.f.
is F(t) 1 - 1 - exp(-tl)n
4
Question 2
Let X(t) t ? 0 be a Poisson process of rate
?. Suppose it is known that X(t) n. For n 1,
2, , determine the mean of the first arrival
time W1 and Wn. Note The Wn here is
conditioned on X(t) n. This is different from
tutorial 9.
5
Question 2
Let Y1, , Yn be IID and uniformly distributed
on (0, t. Then W1 has the same distribution as
Y(1) and Wn has the same distribution as Y(n).
Note Yi is a random variable that represents
the position of customer i, on the time axis
(measured w.r.t t 0), after placing the
customer on the axis (uniform distribution). It
is not necessary the ith arrived customer. By
definition, Y(1) min Y1, Y2, , Yn, that is,
it is the first arrived customer.
6
Question 2
Then W1 has the same distribution as Y(1) and Wn
has the same distribution as Y(n) (since it is
the maximum of all Yi it represents the last
arrived customer). The distribution of Y(1) is
7
Question 2
8
Question 2
9
Question 3
Let X(t) t ? 0 be a Poisson process of rate
?. Suppose it is known that X(t) 2. Determine
the mean W1W2, the product of the first two
arrival times
10
Question 3
Let Y1 and Y2 be IID and distributed uniformly on
(0 t. We have E(Y1) E(Y2) t/2 Let Y(1) and
Y(2) be the order statistics of Y1 and Y2. Then
(W1, W2) have the same distribution as (Y(1),
Y(2)). Because Y(1)Y(2) Y1Y2 Thus E(W1W2)
E(Y1Y2) E(Y1)E(Y2) t2/4
11
Question 4
  • Customers arrive at a certain facility according
    to a
  • Poisson process of rate ?. Suppose that it is
    know
  • that five customers arrived in the first t hours.
  • Determine the mean total waiting time
  • EW1 W5 X(t) 5
  • (b) Determine the mean total waiting time
  • EW1 W5 W6 X(t) 5

12
Question 4
Let Y1, , Y5 be IID and distributed uniformly on
(0, t. We have E(Y1) E(Y5) t/2 Let
Y(1), , Y(5) be the order statistics. Then (W1,
,W5) have the same distribution as (Y(1), ,
Y(5)). Because Y(1) Y(5) Y1 Y5
13
Question 4a
Thus E(W1 W5 X(t) 5) E(Y1 Y5)
E(Y1) E(Y5) 5t/2
14
Question 4b
EW1 W5 W6X(t) 5 EW1 W5 X(t)
5 E W6 X(t) 5 5t/2 (t 1/?) (See
lecture notes for the calculation of EW6 X(t)
5 )
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