The cost of ordering y units of product is a specified function c(y) and it takes l units
time until order is delivered. In addition, the shop pays an inventory holding cost of h
per unit item per b unit time. Whenever customers demands more of the product than
is presently available, than the amount on hand is sold and the remainder of the order
is order is lost to the shop. Define variables and events to analyze this model and
estimate the shop’s expected profit up to fixed time T.
(10 marks)
6. Suppose that the different policyholders of a casually insurance company generate
claims according to independent Poisson process with a common rate
λ
, and that each
claim amount has distribution F. Suppose also that new customers sign up according
to a Poisson process with rate v, and each existing policyholder pays the insurance
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View Full Documentfirm at a fixed rate C per unit time. This system states with n customer and initial
capital a
≥
0, find then. Define the variable and enters for this system and estimate the
probability that the firm’s capital is always non negative at all times up to time T.
(20 marks)
7. A system needs n working machines to be operational. To guard against machine
breakdown, additional machines are kept available as spares. Whenever a machine
breaks down it is immediately replaced by a spare and is itself sent to the repair
facility, which consists of a single repair person who repair failed machines one at a
time. Once a failed machine has been repaired it becomes available as a spare to be
used when the need arises. All repair times are independent random variables having
the common distribution function G. Each time a machine is put into use the amount
of time it functions before breaking down is a random variable, independent of the
past, having distribution function f. The system is said to “Crash” when a machine
fails and no spares are available. Define variables and events to analyze this system
and write a model in simulating to estimate the time at which the system crashes.
(
2
0
m
a
r
k
s
)
8.
Draw a flow diagram for simulating the Repair model.
(10marks)
9.
The quantity X =
∑
which is the arithmetic average of the n data values, is
called the sample mean. Proved that X, sample mean is an unbiased estimator of
population mean
θ
.
And also find its mean square error.
(10 marks)
10. The quantity
S
2
=
2
1
()
1
i
i
n
X
X
n
=
−
−
∑
is called the sample variance and show that
sample
variance
S
is an unbiased estimator of population variance
σ
.
(10 marks)
11. For any set at number x , ….
., x
prove algebraically that
∑
( x – x )
=
∑
x
 n x
where x =
∑
x
/n .
(10 marks)
12. Write a method for determining when to stop generating new data to estimate a
population mean.
(10 marks)
13. Write a procedure for determining when to stop generating new values to estimate a
probability. (The data values are Berroulli random variables).
(10 marks)
14. If the first three data values are X1=5, X2=14, X3=9, and then find their sample mean
and simple variance.
(10 marks)
15. Suppose we are interested in estimating
θ
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 Winter '09
 a7a
 Flow control, Transmission Control Protocol, transport entity, supplier numbers

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