Jointly Distributed Random Variables
Jointly Distributed Random Variables
Definition
Two random variables X and Y are said to be independent if for
every pair of x and y values,
p(x, y ) = pX (x) · pY (y )
when X and Y are discrete
or
f (x, y ) = fX (x) · fY (y )
when X and Y are continuous
If the above relation is not satisfied for all (x, y ), then X and Y
are said to be dependent.
Jointly Distributed Random Variables
Jointly Distributed Random Variables
Definition
If X1 , X2 , . . . , Xn are all discrete random variables, the joint pmf of
the variables is the function
p(x1 , x2 , . . . , xn ) = P(X1 = x1 , X2 = x2 , . . . , Xn = xn )
If the random variables are continuous, the joint pdf of
X1 , X2 , . . . , Xn is the function f (x1 , x2 , . . . , xn ) such that for any n
intervals [a1 , b1 ], . . . , [an , bn ],
P(a1 ≤ X1 ≤ b1 , . . . an ≤ Xn ≤ bn ) =
Z b1 Z bn
···
f (x1 , . . . , xn )dxn . . . dx1
a1
an
Jointly Distributed Random Variables
Jointly Distributed Random Variables
Definition
The random variables X1 , X2 , . . . , Xn are said to be independent if
for every subset Xi1 , Xi2 , . . . , Xik of the variables (each pair, each
triple, and so on), the joint pmf or pdf of the subset is equal to the
product of the marginal pmf’s or pdf’s.
Jointly Distributed Random Variables
Definition
The random variables X1 , X2 , . . . , Xn are said to be independent if
for every subset Xi1 , Xi2 , . . . , Xik of the variables (each pair, each
triple, and so on), the joint pmf or pdf of the subset is equal to the
product of the marginal pmf’s or pdf’s.
e.g. one way to construct a multinormal distribution is to
take the product of pdf’s of n independent standard normal rv’s:
1 −x12 /2
1 −x22 /2
1 −xn2 /2
√ e
f (x1 , x2 , . . . , xn ) = √ e
··· √ e
2π
2π
2π
1
2
2
2
= √
e −(x1 +x2 +···+xn )/2
( 2π)n
Jointly Distributed Random Variables
Jointly Distributed Random Variables
Definition
Let X and Y be two continuous rv’s with joint pdf f (x, y ) and
marginal Y pdf fY (y ). Then for any Y value y for which
fY (y ) > 0, the conditional probability density function of X
given that Y = y is
fX |Y (x | y ) =
f (x, y )
fY (y )
−∞<x <∞
If X and Y are discrete, then conditional probability mass
function of X given that Y = y is
pX |Y (x | y ) =
p(x, y )
pY (y )
−∞<x <∞
Expectations
Expectations
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
What is the expected total expense for that couple?
Expectations
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
What is the expected total expense for that couple?
Let Z = the total expense for that couple. Then
Z = X +Y.
Expectations
Expectations
p(x, y )
x
12
15
20
12
.05
.05
0
y
15
.05
.10
.20
20
.10
.35
.10
Z =X +Y
Expectations
p(x, y )
x
12
15
20
12
.05
.05
0
y
15
.05
.10
.20
20
.10
.35
.10
Z =X +Y
E (Z ) = .05(12 + 12) + .05(12 + 15) + .10(12 + 20)
+ .05(15 + 12) + .10(15 + 15) + .35(15 + 20)
+ 0(20 + 12) + .20(20 + 15) + .10(20 + 20)
= 33.35
Expectations
Expectations
Definition
Let X and Y be jointly distributed rv’s with pmf p(x, y ) or pdf
f (x, y ) according to whether the variables are discrete or
continuous. Then the expected value of a function h(X , Y ),
denoted by E [h(X , Y )] or µh(X ,Y ) , is given by
(P P
h(x, y ) · p(x, y )
E [h(X , Y )] = R ∞x R y∞
−∞ −∞ h(x, y ) · f (x, y )dxdy
if X and Y are discrete
if X and Y are continuo
Expectations
Expectations
Example (Problem 12)
Two components of a minicomputer have the following joint pdf
for their useful lifetimes X and Y :
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
Expectations
Example (Problem 12)
Two components of a minicomputer have the following joint pdf
for their useful lifetimes X and Y :
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
If the lifetime of the minicomputer is the sum of the lifetimes of
the two components, then what is the expected lifetime of the
minicomputer?
Covariance
Covariance
Covariance
Covariance
Covariance
Covariance
Covariance
Covariance
Definition
The covariance between two rv’s X and Y is
Cov (X , Y ) = E [(X − µX )(Y − µY )]
(P P
y (x − µX )(y − µY )p(x, y )
= R ∞x R ∞
−∞ −∞ (x − µX )(y − µY )f (x, y )dxdy
X , Y discrete
X , Y continuou
Covariance
Definition
The covariance between two rv’s X and Y is
Cov (X , Y ) = E [(X − µX )(Y − µY )]
(P P
y (x − µX )(y − µY )p(x, y )
= R ∞x R ∞
−∞ −∞ (x − µX )(y − µY )f (x, y )dxdy
X , Y discrete
X , Y continuou
Remark: The covariance depends on both the set of possible pairs
and the probabilities.
Covariance
Definition
The covariance between two rv’s X and Y is
Cov (X , Y ) = E [(X − µX )(Y − µY )]
(P P
y (x − µX )(y − µY )p(x, y )
= R ∞x R ∞
−∞ −∞ (x − µX )(y − µY )f (x, y )dxdy
X , Y discrete
X , Y continuou
Remark: The covariance depends on both the set of possible pairs
and the probabilities.
Proposition
Cov (X , Y ) = E (XY ) − µX · µY
Covariance
Covariance
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
Covariance
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
Cov (X , Y ) = E (XY ) − µX · µY = 276.7 − 15.9 · 17.45 = −0.755
Covariance
Covariance
If we change the unit for the previous example from dollar to cent,
then the joint pmf would be
y
1200 1500 2000
p(x, y )
1200 .05
.05
.10
.10
.35
x
1500 .05
0
.20
.10
2000
Covariance
If we change the unit for the previous example from dollar to cent,
then the joint pmf would be
y
1200 1500 2000
p(x, y )
1200 .05
.05
.10
.10
.35
x
1500 .05
0
.20
.10
2000
And correspondingly,
Cov (X , Y ) = E (XY ) − µX · µY = −7550
Covariance
Covariance
Definition
The correlation coefficient of X and Y , denoted by Corr (X , Y ),
ρX ,Y or just ρ is defined by
ρX ,Y =
Cov (X , Y )
σX · σY
Covariance
Definition
The correlation coefficient of X and Y , denoted by Corr (X , Y ),
ρX ,Y or just ρ is defined by
ρX ,Y =
Cov (X , Y )
σX · σY
e.g. for the previous example, the correlation coefficient of X and
Y is
−0.755
ρ=
= −0.09
2.91 · 2.94
Covariance
Covariance
Proposition
1. Corr (aX + b, cY + d) = Corr (X , Y ) if a · c > 0.
2. −1 ≤ Corr (X , Y ) ≤ 1.
3. ρ = 1 or −1 iff Y = aX + b for some a and b with a 6= 0.
4. If X and Y are independent, then ρ = 0. However, ρ = 0
does not imply that X and Y are independent
Statistics and Their Distributions
Statistics and Their Distributions
Assume we are running an online retail store. Some factors may be
of great interest to us. Like the distributions of buyers nation-wide,
the time a customer spend on the website per each visit,
costumers’ satisfactories and etc.
Statistics and Their Distributions
Assume we are running an online retail store. Some factors may be
of great interest to us. Like the distributions of buyers nation-wide,
the time a customer spend on the website per each visit,
costumers’ satisfactories and etc.
For some factors, we may get the data for the “whole population”,
like the time spent per each visit. While for others, we can only
obtain information of a sample, like costumers’ satisfactories.
Statistics and Their Distributions
Assume we are running an online retail store. Some factors may be
of great interest to us. Like the distributions of buyers nation-wide,
the time a customer spend on the website per each visit,
costumers’ satisfactories and etc.
For some factors, we may get the data for the “whole population”,
like the time spent per each visit. While for others, we can only
obtain information of a sample, like costumers’ satisfactories.
If we take into account of the future customers, we are unable to
get the information about the population theoretically. All the
information we are dealing with now is just from a sample.
Statistics and Their Distributions
Statistics and Their Distributions
For example, we have the following data about the time spent per
each visit (in min.) from a sample of size 10:
1
2
3
4
5
time 24 51 12 95 26
6
7
8
9 10
time 5 33 62 31 27
Statistics and Their Distributions
For example, we have the following data about the time spent per
each visit (in min.) from a sample of size 10:
1
2
3
4
5
time 24 51 12 95 26
6
7
8
9 10
time 5 33 62 31 27
Each observation is “random”, i.e. we can not predict the exact
value before we obtain the observation.
Statistics and Their Distributions
For example, we have the following data about the time spent per
each visit (in min.) from a sample of size 10:
1
2
3
4
5
time 24 51 12 95 26
6
7
8
9 10
time 5 33 62 31 27
Each observation is “random”, i.e. we can not predict the exact
value before we obtain the observation.
Therefore we can associate a random variable Xi to the ith
observation.
Statistics and Their Distributions
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
the minimum is 5;
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
the minimum is 5;
the mean is 36.6;
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
the minimum is 5;
the mean is 36.6;
the medain is 29;
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
the minimum is 5;
the mean is 36.6;
the medain is 29;
and the standard deviation is 26.4.
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
the minimum is 5;
the mean is 36.6;
the medain is 29;
and the standard deviation is 26.4.
Sometimes, these characteristics are more interesting to us than
the sample data itself.
Statistics and Their Distributions
Often, we are interested in some overall properties of the sample.
For example, the maximum of the previous sample is 95;
the minimum is 5;
the mean is 36.6;
the medain is 29;
and the standard deviation is 26.4.
Sometimes, these characteristics are more interesting to us than
the sample data itself.
We call these characteristics statistics.
Statistics and Their Distributions
Statistics and Their Distributions
Definition
A statistic is any quantity whose value can be calculated from
sample data.
Statistics and Their Distributions
Definition
A statistic is any quantity whose value can be calculated from
sample data.
Remark:
1. A statistic is a random variable.
Statistics and Their Distributions
Definition
A statistic is any quantity whose value can be calculated from
sample data.
Remark:
1. A statistic is a random variable. The reason is prior to
obtaining data, we are not sure what value of any particular
statistic will result.
Statistics and Their Distributions
Definition
A statistic is any quantity whose value can be calculated from
sample data.
Remark:
1. A statistic is a random variable. The reason is prior to
obtaining data, we are not sure what value of any particular
statistic will result.
We use uppercase letters to denote statistics and lowercase
letter to denote the calculated or observed values of statistics.
Statistics and Their Distributions
Statistics and Their Distributions
Remark:
2. A statistic must be calculated from sample data.
Statistics and Their Distributions
Remark:
2. A statistic must be calculated from sample data.
For example, if in addition to the size 10 sample for the
previous example, we also know that the time spent per each
visit is normally distributed with mean µ and variance σ 2 ,
then neither the population mean µ nor the population
variance σ 2 is a statistic.
Statistics and Their Distributions
Remark:
2. A statistic must be calculated from sample data.
For example, if in addition to the size 10 sample for the
previous example, we also know that the time spent per each
visit is normally distributed with mean µ and variance σ 2 ,
then neither the population mean µ nor the population
variance σ 2 is a statistic.
While the sample mean and the sample variance are two valid
statistics, which will be denoted by X and S 2 , respectively.
Statistics and Their Distributions
Remark:
2. A statistic must be calculated from sample data.
For example, if in addition to the size 10 sample for the
previous example, we also know that the time spent per each
visit is normally distributed with mean µ and variance σ 2 ,
then neither the population mean µ nor the population
variance σ 2 is a statistic.
While the sample mean and the sample variance are two valid
statistics, which will be denoted by X and S 2 , respectively.
3. Any statistic, being a random variable, has a probability
distribution.
The probability distribution of a statistic is referred to as its
sampling distribution.
Statistics and Their Distributions
Remark:
2. A statistic must be calculated from sample data.
For example, if in addition to the size 10 sample for the
previous example, we also know that the time spent per each
visit is normally distributed with mean µ and variance σ 2 ,
then neither the population mean µ nor the population
variance σ 2 is a statistic.
While the sample mean and the sample variance are two valid
statistics, which will be denoted by X and S 2 , respectively.
3. Any statistic, being a random variable, has a probability
distribution.
The probability distribution of a statistic is referred to as its
sampling distribution.
The sampling distribution of a statistic DEPENDS on the
sample size n.
Statistics and Their Distributions
Statistics and Their Distributions
Definition
The random variables X1 , X2 , . . . , Xn are said to form a (simple)
random sample of size n if
1. The Xi s are independent random variables.
2. Every Xi has the same probability distribution.
Statistics and Their Distributions
Definition
The random variables X1 , X2 , . . . , Xn are said to form a (simple)
random sample of size n if
1. The Xi s are independent random variables.
2. Every Xi has the same probability distribution.
In words, X1 , X2 , . . . , Xn forms a random sample if the Xi ’s are
independent and identically distributed (iid).
Statistics and Their Distributions
Statistics and Their Distributions
Remark:
When sampling with replacement or from an infinite (conceptual)
population, the two conditions are satisfied and the result can be
regarded as a random sample.
Statistics and Their Distributions
Remark:
When sampling with replacement or from an infinite (conceptual)
population, the two conditions are satisfied and the result can be
regarded as a random sample.
For sampling WITHOUT replacement from a finite population,
although consecutive observations are not independent and
identically distributed, we can still regard the result as a random
sample if the sample size n is much smaller than the population
size N.
Statistics and Their Distributions
Remark:
When sampling with replacement or from an infinite (conceptual)
population, the two conditions are satisfied and the result can be
regarded as a random sample.
For sampling WITHOUT replacement from a finite population,
although consecutive observations are not independent and
identically distributed, we can still regard the result as a random
sample if the sample size n is much smaller than the population
size N.
In practice, if n/N ≤ .05 (at most .05% of the population is
sampled), we can regard the sample as a random sample.
Statistics and Their Distributions
Statistics and Their Distributions
Deriving Sampling Distributions
Example (Problem 38)
There are two traffic lights on my way to work. Let X1 be the
number of lights at which I must stop, and suppose that the
distribution of X1 is as follows:
x1
0 1 2
µ = 1.1, σ 2 = .49
p(x1 ) .2 .5 .3
Let X2 be the number of lights at which I must stop on the way
home; X2 is independent of X1 . Assume that X2 has the same
distribution as X1 , so that X1 , X2 is a random sample of size n = 2.
Statistics and Their Distributions
Deriving Sampling Distributions
Example (Problem 38)
There are two traffic lights on my way to work. Let X1 be the
number of lights at which I must stop, and suppose that the
distribution of X1 is as follows:
x1
0 1 2
µ = 1.1, σ 2 = .49
p(x1 ) .2 .5 .3
Let X2 be the number of lights at which I must stop on the way
home; X2 is independent of X1 . Assume that X2 has the same
distribution as X1 , so that X1 , X2 is a random sample of size n = 2.
a. Let X = (X1 + X2 )/2. Find the probability distribution of X .
Statistics and Their Distributions
Deriving Sampling Distributions
Example (Problem 38)
There are two traffic lights on my way to work. Let X1 be the
number of lights at which I must stop, and suppose that the
distribution of X1 is as follows:
x1
0 1 2
µ = 1.1, σ 2 = .49
p(x1 ) .2 .5 .3
Let X2 be the number of lights at which I must stop on the way
home; X2 is independent of X1 . Assume that X2 has the same
distribution as X1 , so that X1 , X2 is a random sample of size n = 2.
a. Let X = (X1 + X2 )/2. Find the probability distribution of X .
b. Calculate P(X ≤ 1).
Statistics and Their Distributions
Deriving Sampling Distributions
Example (Problem 38)
There are two traffic lights on my way to work. Let X1 be the
number of lights at which I must stop, and suppose that the
distribution of X1 is as follows:
x1
0 1 2
µ = 1.1, σ 2 = .49
p(x1 ) .2 .5 .3
Let X2 be the number of lights at which I must stop on the way
home; X2 is independent of X1 . Assume that X2 has the same
distribution as X1 , so that X1 , X2 is a random sample of size n = 2.
a. Let X = (X1 + X2 )/2. Find the probability distribution of X .
b. Calculate P(X ≤ 1).
c. Calculate µX . How does it relate to µ, the population mean?
Statistics and Their Distributions
Deriving Sampling Distributions
Example (Problem 38)
There are two traffic lights on my way to work. Let X1 be the
number of lights at which I must stop, and suppose that the
distribution of X1 is as follows:
x1
0 1 2
µ = 1.1, σ 2 = .49
p(x1 ) .2 .5 .3
Let X2 be the number of lights at which I must stop on the way
home; X2 is independent of X1 . Assume that X2 has the same
distribution as X1 , so that X1 , X2 is a random sample of size n = 2.
a. Let X = (X1 + X2 )/2. Find the probability distribution of X .
b. Calculate P(X ≤ 1).
c. Calculate µX . How does it relate to µ, the population mean?
d. Calculate σ 2 . How does it relate to σ 2 , the population
X
variance?
Statistics and Their Distributions
Statistics and Their Distributions
Deriving Sampling Distributions
Example
A certain system consists of two identical components. The life
time of each component is supposed to have an expentional
distribution with parameter λ. The system will work if at least one
component works properly and the two components are assumed
to work independently. Let X1 and X2 be the lifetime of the two
components, respectively. What can we say about the lifetime of
the system T0 = X1 + X2 ?
Statistics and Their Distributions
Statistics and Their Distributions
Deriving Sampling Distributions
Example
A certain system consists of two identical components. The life
time of each component is supposed to have an expentional
distribution with parameter λ = 3. The system will work if both
components work properly and the two components are assumed
to work independently. Let X1 and X2 be the lifetime of the two
components, respectively. Then the lifetime of the system is
T1 = min(X1 , X2 ). What is the average lifetime of 5 such systems?
Statistics and Their Distributions
Deriving Sampling Distributions
Example
A certain system consists of two identical components. The life
time of each component is supposed to have an expentional
distribution with parameter λ = 3. The system will work if both
components work properly and the two components are assumed
to work independently. Let X1 and X2 be the lifetime of the two
components, respectively. Then the lifetime of the system is
T1 = min(X1 , X2 ). What is the average lifetime of 5 such systems?
This time, direct derivation of the sampling distribution is
complicated.
Statistics and Their Distributions
Deriving Sampling Distributions
Example
A certain system consists of two identical components. The life
time of each component is supposed to have an expentional
distribution with parameter λ = 3. The system will work if both
components work properly and the two components are assumed
to work independently. Let X1 and X2 be the lifetime of the two
components, respectively. Then the lifetime of the system is
T1 = min(X1 , X2 ). What is the average lifetime of 5 such systems?
This time, direct derivation of the sampling distribution is
complicated.
Instead, we use the method simulation.
Statistics and Their Distributions
Statistics and Their Distributions
Simulation Experiments
1. Use some software to generate a size-5 random sample whose
distribution is EXP(3);
Statistics and Their Distributions
Simulation Experiments
1. Use some software to generate a size-5 random sample whose
distribution is EXP(3);
2. Generate another size-5 random sample whose distribution is
EXP(3);
Statistics and Their Distributions
Simulation Experiments
1. Use some software to generate a size-5 random sample whose
distribution is EXP(3);
2. Generate another size-5 random sample whose distribution is
EXP(3);
3. Construct the data set min(Xi , Yi ) for i = 1, . . . , 5 from these
two random samples;
Statistics and Their Distributions
Simulation Experiments
1. Use some software to generate a size-5 random sample whose
distribution is EXP(3);
2. Generate another size-5 random sample whose distribution is
EXP(3);
3. Construct the data set min(Xi , Yi ) for i = 1, . . . , 5 from these
two random samples;
4. Calculate the mean of the data set. This is one simulation.
Statistics and Their Distributions
Simulation Experiments
1. Use some software to generate a size-5 random sample whose
distribution is EXP(3);
2. Generate another size-5 random sample whose distribution is
EXP(3);
3. Construct the data set min(Xi , Yi ) for i = 1, . . . , 5 from these
two random samples;
4. Calculate the mean of the data set. This is one simulation.
5. Simulate another 499 times;
Statistics and Their Distributions
Simulation Experiments
1. Use some software to generate a size-5 random sample whose
distribution is EXP(3);
2. Generate another size-5 random sample whose distribution is
EXP(3);
3. Construct the data set min(Xi , Yi ) for i = 1, . . . , 5 from these
two random samples;
4. Calculate the mean of the data set. This is one simulation.
5. Simulate another 499 times;
6. Construct the histogram for the 500 results from simulations.
Statistics and Their Distributions
Statistics and Their Distributions
Statistics and Their Distributions
Statistics and Their Distributions
Statistics and Their Distributions
The larger the sample size is, the smaller the spread of the
sampling distribution of the sample mean is.
Statistics and Their Distributions
Statistics and Their Distributions
Example (Problem 45)
Carry out a simulation experiment using a statistical computer
package or other software to study the sampling distribution of X
when the population distribution is lognormal with E (ln(X )) = 3
and V (ln(X )) = 1. Consider the four sample sizes n = 10, 20, 30,
and 50, and in each case use 500 replications.
Statistics and Their Distributions
Example (Problem 45)
Carry out a simulation experiment using a statistical computer
package or other software to study the sampling distribution of X
when the population distribution is lognormal with E (ln(X )) = 3
and V (ln(X )) = 1. Consider the four sample sizes n = 10, 20, 30,
and 50, and in each case use 500 replications.
1. Use some software to generate a size-10 random sample
whose distribution is LOGN(3,1);
Statistics and Their Distributions
Example (Problem 45)
Carry out a simulation experiment using a statistical computer
package or other software to study the sampling distribution of X
when the population distribution is lognormal with E (ln(X )) = 3
and V (ln(X )) = 1. Consider the four sample sizes n = 10, 20, 30,
and 50, and in each case use 500 replications.
1. Use some software to generate a size-10 random sample
whose distribution is LOGN(3,1);
2. Calculate the mean of the random sample. This is one
simulation.
Statistics and Their Distributions
Example (Problem 45)
Carry out a simulation experiment using a statistical computer
package or other software to study the sampling distribution of X
when the population distribution is lognormal with E (ln(X )) = 3
and V (ln(X )) = 1. Consider the four sample sizes n = 10, 20, 30,
and 50, and in each case use 500 replications.
1. Use some software to generate a size-10 random sample
whose distribution is LOGN(3,1);
2. Calculate the mean of the random sample. This is one
simulation.
3. Simulate another 499 times;
Statistics and Their Distributions
Example (Problem 45)
Carry out a simulation experiment using a statistical computer
package or other software to study the sampling distribution of X
when the population distribution is lognormal with E (ln(X )) = 3
and V (ln(X )) = 1. Consider the four sample sizes n = 10, 20, 30,
and 50, and in each case use 500 replications.
1. Use some software to generate a size-10 random sample
whose distribution is LOGN(3,1);
2. Calculate the mean of the random sample. This is one
simulation.
3. Simulate another 499 times;
4. Construct the histogram for the 500 results from simulations.
Statistics and Their Distributions
Example (Problem 45)
Carry out a simulation experiment using a statistical computer
package or other software to study the sampling distribution of X
when the population distribution is lognormal with E (ln(X )) = 3
and V (ln(X )) = 1. Consider the four sample sizes n = 10, 20, 30,
and 50, and in each case use 500 replications.
1. Use some software to generate a size-10 random sample
whose distribution is LOGN(3,1);
2. Calculate the mean of the random sample. This is one
simulation.
3. Simulate another 499 times;
4. Construct the histogram for the 500 results from simulations.
5. Repeat the simulation for n = 20, 30 and 50.
Statistics and Their Distributions
Statistics and Their Distributions
Statistics and Their Distributions
Statistics and Their Distributions
Statistics and Their Distributions
As the sample size becomes larger, the sampling distribution looks
more like the normal distribution.
Statistics and Their Distributions
Statistics and Their Distributions