Chapter 10
Simulation:
An Introduction
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10.1
1. Introduction
1.1 Definition
• A simulation is an imitation of some real things, state of
affairs, or processes.
• The act of simulating something generally entails certain
key characteristics or behaviours of a selected physical or
abstract system.
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10.2
1.2 Use of Simulation
• Simulation is used in many contexts, including the
modeling of natural systems or human systems in order to
gain insight into their functioning.
• Simulation can be used to show the eventual real effects of
alternative conditions and courses of action.
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10.3
Example 1: Checking distribution theory
Theory:
• A sample of size 4 is taken from a normal distribution with
mean 0.
• Consider the statistic
𝑋ത
𝑇=
𝑆/ 4
where 𝑋ത and S are the sample mean and s.d. respectively
• The statistic T follows a t-distribution with 3 degrees of
freedom.
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10.4
Example 1
(Continued)
Simulation:
• Generate 1000 random samples each of size 4 from a
normal distribution.
• For each sample, compute the value of the statistic T.
• Construct a histogram for these 1000 realizations of the
statistic T.
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10.5
Example 1
(Continued)
Result:
• The histogram matches
well with the t(3)
distribution.
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10.6
Example 2: Comparing estimators
We have learnt several robust estimators of location
Question: Which estimator is the best, the trimmed mean or
the Winsorized mean?
Secondary questions:
• Under what conditions ( in terms of the underlying
distributions ) is the estimator better?
We may consider various underlying distributions
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10.7
Example 2: Comparing estimators
• How to measure the performance?
We may use the Mean Squared Error (MSE)
𝑀𝑆𝐸 = 𝐵𝑖𝑎𝑠 2 + 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
• It may be intractable to compute the MSE for each of the
estimators under different underlying distributions.
• Simulation is an alternative solution
• How to design such a simulation study?
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10.8
Example 3: Buffon’s needle experiment
A needle of length 𝑙 is thrown randomly onto a grid of
parallel lines with distance 𝑑 ( > 𝑙 ).
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10.9
Example 3
(Continued)
•
What is the probability that the needle intersects a line?
Answer:
2/𝑙
𝜋𝑑
•
•
Can we get the answer through simulations?
A website gives the visualization of the experiment
http://www.metablake.com/pi.swf
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10.10
Example 3
(Continued)
The experiment
• Simulate the throwing of a needle into a grid of parallel
lines, say N times
• Count the number of times the needle intersects a line, say
n times
• Then n/N gives estimate of the probability that the needle
intersects a line.
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10.11
2. How to do the simulation
• Step 1: Generate the position and the inclination of the
needle.
– Generate a random number, 𝑥, from Uniform (0, 𝑑/2). This
number represents the location of the centre of the needle.
– Generate a random number, 𝜃, from Uniform (0, 𝜋). This
number represents the angle of between the needle and the
parallel lines.
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10.12
How to do the simulation
(Continued)
• Step 2: Check if the needle cuts a line.
– The needle cuts a line if 𝑥 < (𝑙/2) sin(𝜃).
– Create a variable 𝑤 = 1 if 𝑥 < (𝑙/2) sin(𝜃) and 0 otherwise.
• Step 3: Repeat Steps 1 and 2 N times.
– Count the number of times that 𝑤 = 1, let say n times.
– Then an estimate of the probability of the needle intersects a
line is given by 𝑛/𝑁.
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10.13
Example 3: R Code
> # Buffon's needle
> # X ~ U(0,d/2), t ~ U(0,pi) where d is the distance
between 2 parallel lines
> # A needle of length L cut one of the lines if x <
L/2*sin(t)
> # Theoretical Probability = 2*L/(pi*d)
> ns <- 50000
> d <- 2
> L <- 1
> d2 <- d/2
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10.14
Example 3: R Code
> # Buffon's needle
> # Theoretical answer
> 2*L/(pi*d)
[1] 0.3183099
> x <- runif(ns,0,d2) # Generate a sample of size ns
from uniform(0, d2)
> t <- runif(ns,0,pi) # Generate a sample of size ns
from uniform (0, pi)
> length(x[x < L/2*sin(t)])/ns
[1] 0.3186
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10.15
3. Random number generator
3.1 Definition
• A sequence of pseudo-random numbers {Ui} is a
deterministic sequence of number in [0, 1] having the
same relevant statistical properties as a sequence of
random numbers
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3.2 Congruential generators
• Congruential generators are defined by
𝑋𝑖 = 𝑎𝑋𝑖−1 + 𝑐 𝑚𝑜𝑑 𝑀
for a multiplier a, shift c, and modulus M.
• 𝑎, 𝑐 and 𝑀 are all integers
• For example: Let 𝑀 = 10, 𝑎 = 2, 𝑐 = 0, 𝑥0 = 1
• Then the sequence generated is 1, 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, ⋯
• Are these numbers random? There is a pattern, 2, 4, 8 ,6.
• Uniform random numbers are obtained by
𝑈𝑖 = 𝑋𝑖 /𝑀
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3.2 Congruential generators
• To initialize, we have to provide a seed, 𝑋0 .
• If 𝑐 = 0, generators having the form
𝑋𝑖 = 𝑎𝑋𝑖−1 𝑚𝑜𝑑 𝑀
are called multiplicative congruential generator.
• If 𝑐 > 0, they are called linear congruential generator.
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10.18
Remarks
• M + 1 values {𝑋0 , 𝑋1 , ⋯ , 𝑋𝑀 } cannot be distinct and at least one
value must occur twice, as 𝑋𝑖 and 𝑋𝑖+𝑘 , say.
• 𝑋𝑖 , 𝑋𝑖+1 , ⋯ , 𝑋𝑖+𝑘−1 is repeated as 𝑋𝑖+𝑘 , 𝑋𝑖+𝑘+1 , ⋯ , 𝑋𝑖+2𝑘−1
• The sequence {𝑋𝑖 } is periodic with period 𝑘 ≤ 𝑀
• For multiplicative generators, the maximal period is 𝑀 − 1
If 0 ever occurs, it is repeated indefinitely.
• One of our primary objectives is to use a generator with as
large period as possible.
• However, a large period does not guarantee a good generator.
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10.19
3.3 R function for congruential generators
> # Linear congruential generators
> lcg <- function(n,a,m,c,x0){
+
ran <- NULL
+
for (i in 1:n) {
+
x1 <- (a*x0+c)%%m
+
x0 <- x1
+
ran <- c(ran,x1/m) }
+ return(ran) }
> lcg(10,397204094,2^31-1,0,1234)
[1] 0.24381116
0.08947511
0.38319371
0.72792771
0.17503827
[7] 0.40680994
0.90923700
0.34271140
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0.72800325
0.55960111
10.20
4. Generate uniform random numbers
4.1 SAS code:
*Generate Uniform Random Numbers;
data try1;
seed = 1234;
do i = 1 to 10;
x = ranuni(seed);
output;
end;
keep x;
run;
proc print data=try1;
var x;
run;
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10.21
4. Generate uniform random numbers
4.1 SAS code (Continued):
* Alternative way to generate Uniform Random Numbers;
data try1;
seed = 1234; call streaminit(seed);
do i = 1 to 10;
x = rand(‘uniform’); output;
end; keep x;
run;
• Without call streaminit(seed), the sets of random numbers
generated will be different.
• If seed value is not specified, then the time at which the
command is executed is used for the seed.
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10.22
Generate uniform random numbers
(Continued)
4.2 R code
> # Generate 1000 random numbers from U(0, 1) distribution
> set.seed(1234)
> x <- runif(1000)
In general, “runif(n, a, b)” generates a vector of n random
numbers from a uniform distribution between a and b.
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10.23
5 Generate non-uniform random numbers
5.1 Inversion method
• If X has a continuous distribution function 𝐹(𝑥) (Recall
𝐹 𝑥 = Pr(𝑋 ≤ 𝑥)), then
𝐹 𝑥 ~ 𝑈𝑛𝑖𝑓𝑜𝑟𝑚(0, 1)
Algorithm:
• Generate U from Uniform(0, 1)
• Set 𝑋 = 𝐹 −1 (𝑈) provided that the inverse function exists.
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10.24
5.2 Exponential distribution
• If X follows an exponential distribution with parameter λ
(i.e. 𝐸 𝑋 = 𝜆), then
𝑥
1 −𝑡
𝐹 𝑥 = Pr 𝑋 ≤ 𝑥 = න 𝑒 𝜆 𝑑𝑡
0 𝜆
• Note : The p.d.f. of 𝑋 is given by 𝑓 𝑡 =
1 −𝑡
𝑒 𝜆
𝜆
for 0 ≤ 𝑡 ≤ ∞
• Let 𝑦 = 𝑡/𝜆. Then 𝑑𝑦 = 𝑑𝑡/𝜆. 𝑡 = 0 ⇒ 𝑦 = 0 and 𝑡 = 𝑥 ⇒
𝑦 = 𝑥/𝜆. Hence
𝑥/𝜆
𝐹(𝑥) = න 𝑒 −𝑦 𝑑𝑦 = 1 − 𝑒 −𝑥/𝜆
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10.25
5.2 Exponential distribution
• Let 𝑈 = 𝐹 𝑋 = 1 − 𝑒 −𝑋/𝜆 .
• Then 𝑋 = 𝐹 −1 𝑈 = −𝜆 log(1 − 𝑈)
Suppose 𝜆 = 2.
Step 1: Generate U from Uniform(0, 1)
Step 2: Set 𝑋 = −2 log(1 − 𝑈).
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10.26
5.3 Weibull distribution
If X follows a Weibull distribution with parameter 𝛽, then it
can be shown that
𝐹 𝑥 = 1 − exp −𝑥 𝛽 , on (0, ∞)
• Note: The p.d.f. for 𝑋 is given by 𝑓 𝑥 = 𝛽𝑥 𝛽−1 exp −𝑥 𝛽 ,
for 𝑥 in 0, ∞
1. Generate U from Uniform(0, 1)
2. Set 𝑋 = − log 1 − 𝑈 1/𝛽 .
(It is obtained by solving for 𝑋 in the equation 𝑈 = 1 − exp(−𝑥 𝛽 ))
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10.27
5.4 Cauchy distribution
• If 𝑋 follows a Cauchy distribution with parameter 𝜇 and 𝜎,
then it can be shown that
1 1
𝑥−𝜇
−1
𝐹 𝑥 = + tan
,
for 𝑥 in (−∞, ∞)
2 𝜋
𝜎
• Note:
1
𝑓 𝑥 =
𝑥−𝜇 2
𝜋𝜎 1 +
𝜎
1. Generate U from Uniform(0, 1).
2. Set 𝑋 = 𝜎 tan 𝜋 𝑈 − 0.5 + 𝜇
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10.28
6. Algorithm to generate a normal random variable
6.1 Box-Muller Algorithm
• Generate 𝑈1 and 𝑈2 from Uniform(0, 1).
• Set 𝜃 = 2𝜋𝑈1 .
• Set 𝑅 = −2 log 𝑈 1/2
2
• Set 𝑋 = 𝑅 cos(𝜃) and 𝑌 = 𝑅 sin 𝜃 .
• X and Y are independent standard normal variables.
• For simplicity, only X or Y is used.
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10.29
7. Generate a random variable from other
random variables
7.1 Cauchy distribution
• If Y and Z are independent and follow 𝑁(0, 1), then
𝑋 = 𝑌/𝑍
follows a Cauchy(0, 1) distribution.
• If 𝑌~𝑁(𝜇, 𝜎 2 ) and 𝑍~𝑁(0, 1), and are independent, then
𝑋 = 𝑌/𝑍
follows a Cauchy(𝝁, 𝝈𝟐) distribution
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10.30
Generate a random variable from other random
variables (Continued)
7.2 Chi-square distribution
• If 𝑌~𝑁(0, 1) and 𝑋 = 𝑌 2 , then
𝑋 ~𝜒 2 (1)
• If 𝑌1 , 𝑌2 , ⋯ , 𝑌𝑛 are independent and identically distributed
standard normal variables and 𝑋 = 𝑌12 + 𝑌22 + ⋯ + 𝑌𝑛2 ,
then
𝑋~𝜒 2 (𝑛)
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Generate a random variable from other random
variables (Continued)
7.3 Student’s t-distribution
• If 𝑌 ~ 𝑁(0, 1) and 𝑍 ~ 𝜒 2 (𝑝), then
𝑌
𝑋=
𝑍/𝑝
follows a Student’s t distribution with p degrees of
freedom
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10.32
Generate a random variable from other random
variables (Continued)
7.4 F distribution
• If 𝑌 ~ 𝜒 2 𝑚 , and 𝑍 ~ 𝜒 2 𝑛 , then
𝑌/𝑚
𝑋=
𝑍/𝑛
follows an F distribution with degrees of freedom m and n.
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10.33
8. Functions to generate random numbers from
specific distributions
References
SAS
http://support.sas.com/documentation/cdl/en/lrdict/6431
6/HTML/default/viewer.htm#a001466748.htm
R
https://stat.ethz.ch/R-manual/Rdevel/library/stats/html/Distributions.html
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10.34
8.1 Function to generate random numbers from
Uniform (a, b)
To generate random numbers from Uniform (a, b)
1
𝑓 𝑥 =
,
for 𝑎 < 𝑥 < 𝑏
𝑏−𝑎
R program
>
>
>
>
>
>
>
# Generate uniform random variables
n <- 100
a <- 0
b <- 100
set.seed(1234) # set the seed number
x <- runif(n,a,b)
x
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10.35
8.1 Function to generate uniform distribution
random variables (Continued)
SAS program
/* Generate uniform random variables */
data unif;
call streaminit(1234); /* 1234 is used as the seed */
n = 100; a = 0; b = 10;
do i = 1 to n;
x = a + (b - a)*rand('uniform');
output;
end;
keep x;
proc print data=unif;
run;
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10.36
8.2 Function to generate Normal distribution
random variables
To generate random numbers from Normal(𝜇, 𝜎2)
1
𝑥−𝜇 2
𝑓 𝑥 =
exp −
,
for − ∞ < 𝑥 < ∞
2
2𝜎
2πσ
R program
>
>
>
>
>
>
# Generate normal random variables
n <- 100
mu <- 0
sigma <- 1
x <- rnorm(n, mean=mu, sd=sigma)
x
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10.37
8.2 Function to generate Normal distribution
random variables (Continued)
SAS program
/* Generate normal random variables */
data norm;
seed = 1234;
call streaminit(seed);
n = 100; mu = 0; sigma = 1;
do i = 1 to n;
x = rand('normal',mu,sigma);
output;
end;
keep x;
run;
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10.38
8.3 Function to generate Exponential distribution
random variables
To generate random numbers from Exponential(λ)
1
𝑥
𝑓 𝑥 = exp − ,
𝜆
𝜆
for 𝑥 > 0
R program
>
>
>
>
>
# Generate exponential random variables
n <- 100
lambda <- 5
x <- rexp(n, mean=lambda)
x
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8.3 Function to generate Exponential distribution
random variables (Continued)
SAS program
/* Generate exponential random variables */
data expno;
seed = 1234;
call streaminit(seed);
n = 100; lambda = 5;
do i = 1 to n;
x = lambda*rand('exponential');
output;
end;
keep x;
run;
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10.40
8.4 Function to generate Gamma distribution
random variables
To generate random numbers from Gamma(𝛼, 𝛽)
1
𝑥
𝛼−1
𝑓 𝑥 = 𝛼
𝑥
exp −
,
𝛽 Γ 𝛼
𝛽
for 𝑥 > 0
R program
>
>
>
>
>
>
# Generate Gamma random varaibles
n <- 100
alpha <- 1
beta <- 2
x <- rgamma(n, shape = alpha, scale = beta)
x
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8.4 Function to generate Gamma distribution
random variables (Continued)
SAS program
/* Generate Gamma random variables */
data gammano;
seed = 1234;
call streaminit(seed);
n = 100; alpha = 1; beta = 2;
do i = 1 to n;
x = beta*rand('gamma',alpha);
output;
end;
keep x;
run;
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10.42
8.5 Function to generate Chi-square distribution
random variables
To generate random numbers from Chi-square(p)
𝑓 𝑥 =
1
𝑝
22 Γ
𝑝
2
𝑝
𝑥 2−1 exp
𝑥
− ,
2
for 𝑥 > 0
R program
>
>
>
>
>
#
n
p
x
x
Generate Chisquare random variables
<- 100
<- 10
<- rchisq(n, df = p)
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8.5 Function to generate Chi-square distribution
random variables (Continued)
SAS program
/* Generate chisquare random variables */
data chisqno;
seed = 1234;
call streaminit(seed);
n = 100; df = 10;
do i = 1 to n;
x = rand('chisquare',df);
output;
end;
keep x;
run;
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10.44
8.6 Function to generate Beta distribution
random variables
To generate random numbers from Beta(α, β)
Γ 𝛼 + 𝛽 𝛼−1
𝑓 𝑥 =
𝑥
1−𝑥
Γ 𝛼 Γ 𝛽
𝛽−1 ,
for 0 < 𝑥 < 1
R program
>
>
>
>
>
>
#
n
a
b
x
x
Generate Beta random variables
<- 100
<- 2
<- 3
<- rbeta(n, shape1 = a, shape2 = b)
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8.6 Function to generate Beta distribution
random variables (Continued)
SAS program
/* Generate Beta random variables */
data betano;
seed = 1234;
call streaminit(seed);
n = 100; alpha = 2; beta = 3;
do i = 1 to n;
x = rand('beta',alpha,beta);
output;
end;
keep x;
run;
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10.46
8.7 Function to generate t-distribution random
variables
To generate random numbers from t(k)
𝑘+1
Γ
2
𝑓 𝑥 =
𝑘
Γ
2
1
1
𝑘𝜋
1+
𝑘+1 ,
𝑥2 2
for − ∞ < 𝑥 < ∞
𝑘
R program
>
>
>
>
>
#
n
k
x
x
Generate t random variables
<- 100
<- 5
<- rt(n, df = k)
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10.47
8.7 Function to generate t-distribution random
variables (Continued)
SAS program
/* Generate t random variables */
data tno;
seed = 1234;
call streaminit(seed);
n = 100; df = 5;
do i = 1 to n;
x = rand('t',df);
output;
end;
keep x;
run;
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10.48
8.8 Function to generate F-distribution random
variables
To generate random numbers from F(m, n)
𝑛1 + 𝑛2
Γ
𝑓 𝑥 = 𝑛 2 𝑛
Γ 1 Γ 2
2
2
𝑛1
𝑛2
𝑛1
2
𝑛1
𝑥 2 −1
𝑛1
1+ 𝑥
𝑛2
𝑛1 +𝑛2
2
,
for 0 < 𝑥 < ∞
R program
>
>
>
>
>
>
# Generate F random variables
n <- 100
n1 <- 5
n2 <- 10
x <- rf(n, df1 = n1, df2 = n2)
x
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8.8 Function to generate F-distribution random
variables (Continued)
SAS program
/* Generate F random variables */
data fno;
seed = 1234;
call streaminit(seed);
n = 100; df1 = 5; df2 = 10;
do i = 1 to n;
x = rand('f',df1,df2);
output;
end;
keep x;
run;
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8.9 Function to generate Binomial distribution
random variables
To generate random numbers from Binomial(n, p)
𝑛 𝑥
𝑓 𝑥 =
𝑝 1−𝑝
𝑥
𝑛−𝑥 ,
for 𝑥 = 0, 1, ⋯ , 𝑛
R program
>
>
>
>
>
>
# Generate Binomial random variables
nn <- 100
n <- 10
p <- 0.3
x <- rbinom(100, size = n, prob = p)
x
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8.9 Function to generate Binomial distribution
random variables (Continued)
SAS program
/* Generate Binomial random variables */
data binomno;
seed = 1234;
call streaminit(seed);
ns = 100; n = 10; p = 0.3;
do i = 1 to ns;
x = rand('binomial',p,n);
output;
end;
keep x;
run;
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8.10 Function to generate Poisson distribution
random variables
To generate random numbers from Poisson(λ)
𝑒 −𝜆 𝜆𝑥
𝑓 𝑥 =
,
𝑥!
for 𝑥 = 0, 1, 2, ⋯
R porgram
>
>
>
>
>
# Generate Poisson random variables
n <- 100
lambda <- 3
x <- rpois(100, lambda)
x
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8.10 Function to generate Poisson distribution
random variables (Continued)
SAS program
/* Generate Poisson random variables */
data poisno;
seed = 1234;
call streaminit(seed);
n = 100; lambda = 3;
do i = 1 to n;
x = rand('poisson',lambda);
output;
end;
keep x;
run;
ST2137 Computer Aided Data Analysis
CYM
10.54
8.11 Function to generate Hypergeometric
distribution random variables
To generate random numbers from Hypergeo(n, N, S)
𝑆
𝑓 𝑥 = 𝑥
R program
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𝑁−𝑆
𝑛−𝑥 ,
𝑁
𝑛
for 𝑥 = 0, 1, 2, ⋯ , min(𝑛, 𝑆)
# Generate Hypergeometric random variables
ns <- 100
n <- 10
S <- 20
N <- 50
x <- rhyper(ns,S,N,n)
x
ST2137 Computer Aided Data Analysis
CYM
10.55
8.11 Function to generate Hypergeometric
distribution random variables (Continued)
SAS porgram
/* Generate Hypergeometric random variables */
data hypergeono;
seed = 1234;
call streaminit(seed);
ns = 100; popnsize = 10; numbsucc = 5; samplsize =3;
do i = 1 to ns;
x=rand('hyper',popnsize,numbsucc,samplsize);
output;
end;
keep x;
run;
ST2137 Computer Aided Data Analysis
CYM
10.56
8.12 Function to generate Negative Binomial
distribution random variables
To generate random numbers from NBinom(r, p)
𝑟+𝑥−1 𝑟
𝑓 𝑥 =
𝑝 1 − 𝑝 𝑥,
𝑥
for 𝑥 = 0, 1, 2, ⋯
R program
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#
n
r
p
x
x
Generate Negative Binomial random var
<- 100
<- 10
<- 0.3
<- rnbinom(n, size=r, prob=p)
ST2137 Computer Aided Data Analysis
CYM
10.57
8.12 Function to generate Negative Binomial
random variables (Continued)
SAS program
/* Generate Negative Binomial random variables */
data negbinomno;
seed = 1234;
call streaminit(seed);
n = 100; p=0.3; k=5;
do i = 1 to n;
x = rand('negbinomial',p,k);
output;
end;
keep x;
run;
ST2137 Computer Aided Data Analysis
CYM
10.58