MATH268: Assignment 5
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MATH268: Assignment 5
Exercise 1 Let the random variable TS have the p . d.f. f given by
f (t) = 0 if t < 
and
f (t) = 
if t > 
.
(a) Obtain the c. d.f. F .
[marks 5]
(b) Using the inverse transformation method, derive a formula for simulating TS . If the
inverse transformation method produces a formula in terms of 1 _ u with u being a U(0, 1) random number, replace it with u . This makes computations easier.
[marks 10]
(c) Let the following be U(0, 1) random numbers:
u1 = 0.67, u2 = 0.7607, u3 = 0.0604,
u4 = 0.1726, u5 = 0.851.
Consider a random variable TA ~ exp(0.5) . Use the formula justified in Example 2. 0. 0. 1 in the lecture notes titled “Simulation” and the above data to simulate TA . These random numbers are denoted by ti(A), i = 1, 2, 3, 4, 5.
[marks 10]
(d) Let the following be U(0, 1) random numbers:
v1 = 0.2942, v2 = 0.0001, v3 = 0.9346,
v4 = 0.5272, v5 = 0.7753.
Use the formula obtained in part (b) and the above data to simulate TS . These random numbers are denoted by ti(S), i = 1, 2, 3, 4, 5.
[marks 10]
(e) Let us understand ti(A), i = 1, 2, 3, 4, 5 (calculated in part (c)) as the inter- arrival times
of an M/G/1 queue, which is initially idle. Assume this queueing system is basic, i. e., the queue is FIFO, and the server can only handle one customer at a time. Let us also understand ti(S), i = 1, 2, 3, 4, 5 (calculated in part (d)) as the corresponding service times. Assume no further arrivals after the time moment i(5)=1 ti(A) . If n(t) denotes the number of customers in the system at time t > 0 corresponding to ti(A), ti(S), i = 1, 2, 3, 4, 5, then draw its graph over the time horizon [0, 12] .
[marks 10]
(f) Calculate the fraction of time on the interval [0, 12] during which the system is idle.
[marks 5]
Exercise 2 Monthly prices of a particular asset are given below:
1 2 3 4 5 6 7 8 9 10 11 12
Month
Price 
121 127 118 120 117 117 119 125 124 128 115 122
(a) Perform the moving average forecasting procedure with n = 3, that is, calculate Ft+1 ,
t = 3, 4, ..., 11.
[marks 10]
(b) Calculate forecasts Ft+1 , t = 1, 2, ..., 11, using exponential smoothing with α = 0.3, A1 =
x1 = 121 .
[marks 10]
Exercise 3 (a) Describe Holt’s method for time series forecast. Write the relevant equations for forecasts.
[marks 10]
(a) Describe Winter’s method for time series forecast. Write the relevant equations for the
forecasts.
[marks 15]
(a) Show how the Winter method reduces to Holt method when the seasonal factors are taken
to be 1 .
[marks 5]
2023-01-03