1.
Suppose that
f
(
x
) = sin x for 0
≤
x
≤
π
/2.
(a) Sketch this probability density function.
(b) Give an acceptance rejection algorithm for generating samples of a random
variable with probability density function
f
.
(c) Use your algorithm and the random numbers in Table 1 to generate 4 samples
from the probability density function
f
.
(d) On average, how many pairs of random numbers are required to generate a
single sample from the probability density function
f
?
(e) What is the probability that 10 or more pairs of random numbers will be
needed to generate a single sample from the probability density function
f
?
2.
You built a LCG with a period of 2
31
. You have just developed a big simulation in
which each replication requires exactly 2
15
random variables. Suppose that you
perform 2
40
replications of your simulation experiment, obtaining the outcomes
X
1
,
X
2
,...,
X
n
, where
n
= 2
40
. Is it reasonable to assume that the
X
i
’s are
uncorrelated? Explain. Would your answer be different if
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 '08
 TOPALOGLU
 Normal Distribution, Variance, Probability theory, probability density function, Cumulative distribution function

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