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Ch5. Probability Densities II
Dr. Deshi Ye
yedeshi@zju.edu.cn
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5.4 Other Prob. Distribution
Uniform distribution: equally likely outcome
else
xforxf
0
1
)(
Mean of uniform2
1
dxx
12
)(
3
1
22
22
222
2
dxx
Variance of uniform
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Ex.
Students believe that they will get the final scores between 80 and 100. Suppose that the final scores given by the instructors has a uniform distribution.
What is the probability that one student get the final score no less than 85?
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Solution
P(85 x 100)= (Base)(Height) = (100 - 85)(0.05) = 0.75
8080 100100
ff((xx))
xx8585
20
1
80100
11
20
1
80100
11
0.050.05
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5.6 The Log-Normal Distr.
Log-Normal distribution:
0
0,02
1)(
2
2
2
)(ln
1
xfordxexxf
x
1,0 It has a long right-hand tail
By letting y=lnx)
ln()
ln(
2
1ln
ln
2
)(2
2
a
Fb
Fdyeb
a
y
,Hence
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Mean of Log-Normal
Mean and variance are
2 2 2/ 2 2 2, ( 1)e e e
Proof.
dxe x2
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Gamma distribution
else
xforexxf
x
0
,0,,0)(
1)(
1
integerpositiveaisαwhen)!1(
)1()1()(0
1
dxex x
2 2, Mean and Variance
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The Exponential Distribution
By letting in the Gamma distribution
10, , 0,
( )
0
x
e for xf x
else
1
2 2, Mean and Variance
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5.8 The Beta Distribution
When a random variables takes on values on the interval [0,1]
else
xforxxxf
0
0,,10)1()()(
)(
)(11
22
,( ) ( 1)
Mean and Variance
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Beta distribution
Are used extensively in Bayesian statistics
Model events which constrained to take place within a interval defined by minimum and maximum value
Extensively used in PERT, CPM, project management
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5.9 Weibull Distribution
0
0,,0)(
1 xforex
xfx
1 22 21 2 1
(1 ), ( (1 ) ( (1 )) )
Mean and Variance
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Weibull distribution
Is most commonly used in life data analysis
Manufactoring and delivery times in industrial engineering
Fading channel modeling in wireless communication
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5.10 Joint distribution
Experiments are conduced where two or more random variables are observed simultaneously in order to determine not only their individual behavior but also the degree of relationship between them.
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Two discrete random variables
),(),( 221121 xXxXPxxf
The probability that X1 takes value x1 and X2 will take
the value x2
EX. x10 1 2
x2
0 1
0.1 0.4 0.10.2 0.2 0
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Marginal probability distributions
2
),()()( 211111xall
xxfxXPxf
x10 1 2
x2 0
1
0.1 0.4 0.10.2 0.2 0
0.3 0.6 0.1)( 11 xf
EX.
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Conditional Probability distribution
0)(),(
)|( 22122
21211 xfprovidedxallfor
xf
xxfxxf
The conditional probability of X1 given that X2=x2
If two random variables are independent
2111211 )()|( xandxallforxfxxf
)(, 221121 xfxfxxf
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EX. With reference to the previous example, find
the conditional probability distribution of X1, given that X2=1. Are X1 and X2 independent?
Solution.
04.0
0
)1(
)1,2()1|2(
5.04.0
2.0
)1(
)1,1()1|1(
5.04.0
2.0
)1(
)1,0()1|0(
21
21
21
f
ff
f
ff
f
ff
)0(3.05.0)1|0( 11 ff Hence, it is dependent
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Continuous variables
If are k continuous random variables, we refer to as the joint probability density of these random variables
1
1
2
22121 ),,,(
b
a
b
a
b
a kk
k
k
dxdxdxxxxf
kXXX ,,, 21
),,,( 21 kxxxf
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EX.
P179.
elsewhere
xxforexxf
xx
0
0,06),( 21
32
21
21
Find the probability that the first random variable between 1 and 2 and the second random variable between 2 and 3
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Marginal density
Marginal density of X1
22111 ),()( dxxxfxf
0 12
3211 06)( 21 xfordxexf xx
Example of previous
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Distribution function
1 2
212121 ),(),(x x
dxdxxxfxxF
),()(
),()(
222
111
xFxF
xFxF
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Independent
),()()(),( 21221121 xxallforxfxfxxf
If two random variables are independent iff the following equation satisfies.
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Properties of Expectation
Consider a function g(x) of a single random variable X. For example: g(x) =9x/5 +32.
If X has probability density f(x), then the mean or expectation of g(x) is given by
dxxfxgxgE )()()]([
[ ( )] ( ) ( )i
i iall x
E g x g x f xOr
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Properties of Expectation
bxaEbaxE ][][
If a and b are constants
][][ 2 xDabaxD
Proof. Both in continuous and discrete case
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Covariance
Covariance of X1 and X2: to measure
)])([( 2211 XXE
Theorem. When X1 and X2 are independent, their covariance is 0
1 1 2 2[( )( )] 0E X X
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5.11 Checking Normal
Question: A data set appears to be generated by a normal distributed random variable
Collect data from students’ last 4 numbers of mobiles
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Simple approach
Histogram can be checked for lack of symmetry
A single long tail certainly contradict the assumption of a normal distribution
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Normal scores plot Also called Q-Q plot, normal quantile plot, normal order plot, or rankit plot.Normal scores: an idealized sample from the
standard normal distribution. It consists of the values of z that divide the axes into equal probability intervals. For example, n=4.
84.0
25.0
25.0
84.0
2.04
4.03
4.02
2.01
zm
zm
zm
zm
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Steps to construct normal score plot
1) order the data from smallest to largest
2) Obtain the normal scores 3) Plot the i-th largest observation,
versus i-th normal score mi, for all i. Plot
nddd 21
),( ii md
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Normal scores in Minitab
In minitab, the normal scores are calculated in different ways:
The i-the normal score is
))4/1/()8/3((1 ni
Where is the inverse cumulative distribution function of the standard normal
)(1 x
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Property of Q-Q plot
If the data set is assumed to be normal distribution, then normal score plot will resemble to a /line through the original.
045
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5.12 Transform observation to near normality
When the histogram or normal scores plot indicate that the assumption of a normal distribution is invalid, transformations of the data can often improve the agreement with normality.
Make larger values smaller
Make large value larger
x
1 ln x 1/ 4x x 2 3,x x
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Simulation
Suppose we need to simulate values from the normal distribution with a specified 2 and
x
z•From The value x can be calculated from the value of a standard normal variable z
1) z can be obtained from the value for a uniform variable u by numerically solving u=F(z)
2) Box-Muller-Marsaglia method: it starts with a pair of independent variable u1 and u2, and produces two standard normal variables
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Box-Muller-Marsaglia
)2sin()ln(2
)2cos()ln(2
122
121
uuz
uuz
22
11
zx
zx
Then
It starts with a pair of independent variable u1 and u2, and produces two standard normal variables
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Simulation from exponential distribution
Suppose we wish to simulate an observation from the exponential distribution
xexF x 0,1)( 3.0
The computer would first produce the value u from the uniform distribution. Then
3.0
)1ln( ux
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Population and sample
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Population and Sample
Investigating: a physical phenomenon, production process, or manufactured unit, share some common characteristics.
Relevant data must be collected. Unit: the source of each measurement.
A single entity, usually an object or person Population: entire collection of units.
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Population and sample
Population
sample
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Key terms
Population All items of interest
Sample Portion of population
Parameter Summary Measure about Population
Statistic Summary Measure about sample
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Examples
Population Unit variables
All students currently enrolled in school
student GPANumber of credits
All books in library
book Replacement cost
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Sample
Statistical population: the set of all measurement corresponding to each unit in the entire population of units about which information is sought.
Sample: A sample from a statistical population is the subset of measurements that are actually collected in the course of investigation.
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Sample
Need to be representative of the population
To be large enough to contain sufficient information to answer the question about the population
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Discussion P10, Review Exercises 1.2 A radio-show host announced that she wanted
to know which singer was the favorite among college students in your school. Listeners were asked to call and name their favorite singer. Identify the population, in terms of preferences, and the sample.
Is the sample likely to be more representative? Comment. Also describe how to obtain a
sample that is likely to be more representative.
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