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
(F) =E[X] by using the sample
mean
1
n
i
i
x
X
n
=
=
∑
. If the observed data are
x
i
, i=1,….n, then the empirical
distribution F
e
puts weight 1/n on each of the points x
1
,…..., x
n
(combining weight of
i=1
n
i
n
X
1
n
n
2
n
i=
1
2
2
i=
1
i=
1
n
i
i
i

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the x
i
are not all distinct). Prove that the bootstrap approximation to the mean square
error MSE(F) is identical to the usual estimate of the mean square error.(10 marks)
16. A service system in which no new customers are allowed to enter after 5 Pm. Suppose
that each day follow the same probability law and estimate the long-sum average
amount of time a customer spends in the system. And then prove that the MSE of
estimate of average time a customer spends in the system is identical to the bootstrap
approximation.
(20 marks)
17. Give a probabilistic proof of
∑
∑
=
=
−
=
−
n
1
i
n
1
i
2
2
i
2
i
nx
x
)
x
(x
where
∑
=
=
n
1
i
i
n
x
x
letting X
denote a random variable that is equally likely to take an any of the values x
1
,…..,x
n
and then by applying the identify Var (X) = E[X ] – (E[X])
2
.
(10 marks)
18
.
To estimate E[X], X1,……,X16 have been simulated with the following values
resulting, 10,11,10.5,11.5,14,8,13,6,15,10,11.5,10.5,12,8,16,5. Based on these data, if
we want the standard deviation of the estimator of E[X] to be less than 0.1, roughly
how many additional simulation runs will be needed?
(10 marks)
19. Suppose we want to use simulation to estimate
1
0
u
x
E e
e dx
θ
⎡
⎤
=
=
⎣
⎦
∫
. Using antithetic
variables, discuss what kind of improvement is possible.
(10 marks)
20. Consider a sequence of random numbers and let N be the first one that is greater than
its immediate predecessor. That is N= min (n: n
≥
2 , U
n
>U
n-1
). Explain what kind of
improvement is possible on the variance of estimator “e” by using antithetic variable.
(20 marks)
21. Suppose we were interested in using simulation to compute
θ
=E [e
u
]. Here, a natural
variant to use as a control is the random number U. Discuss what sort of improvement
over the raw estimator is possible.
(20 marks)