Post on 30-Dec-2015
Chap. 4Continuous Distributions
Examples of Continuous Random Variable
If we randomly pick up a real number between 0 and 1, then we define a continuous uniform random variable V with , for any real number 0≤t≤1.
ttV Prob
Distribution Function
The distribution function of a continuous random variable X is defined as same as that of a discrete random variable, i.e.
).Prob( tXtFX
Probability Density Function
The probability density function( p.d.f.) of a continuous random variable is defined as
For example, the p.d.f. of an uniform random variable defined in [0,2] is
.)(
dt
tdFtf X
X
].2,0[for 2
1 ttf X
The p.d.f. of a continuous random variable with space S satisfies the following properties:
.
1
0
dx xfvent A is ility of eThe probabc
.dxxfb
S .x for all xfa
A X
S X
X
Uniform Distribution
Let random variable X correspond to randomly selecting a number in [a,b]. Then,
b xif 1
bxa if
a xif 0
)(Prob)(b-a
x-axXxFX
Uniform Distribution
otherwise 0
bxa if 1
)()( ab
dx
xdFxf X
X
X has a uniform distribution.
Some Important Observations
The p.d.f. of a continuous random variable does not have to be bounded. For example, the p.d.f. of a uniform random variable with space [0,1/m] is
The p.d.f. of a continuous random variable may not be continuous, as the above example demonstrates.
se otherwi
m
, m if x xf X
0
1
0
Expected Value and Variance
The expected value of a continuous random variable X is
The variance of X is
.dxxxfXE
.2 dxxfxXVar X
Expected Value of a Function of a Random Variable
Let X be a continuous random variable and Y=G(x). Then,
X. of p.d.f. theis )( where
,)()(][
X
X
f
dxxfxGYE
In the following, We will only present the proof for the cases, in which G(.) is a one-to-one monotonic function.
Expected Value of a Function of a Random Variable
dxxfxGdydy
ydFxGYE
dy
dxxf
dy
dx
dx
yGdF
dy
ydF
yGxdydy
ydFxG
dyyyfYE
proof
Y
XY
Y
Y
)()()(
)(][
Therefore,
)())(()(
)( where, )(
)(
)(][
:
1
1
Moment-Generating Function and Characteristic Function
The moment-generating function of a continuous random variable X is
Note that the moment-generating function, if it is finite for -h<t<h for some h>0, completely determines the distribution. In other words, if two continuous random variables have identical m.g.f., then they have identical probability distribution function.
.][
dxxfeeEtM txtX
X
Moment-Generating Function and Characteristic Function
The characteristics function of X is defined to be the Fourier transformation of its p.d.f.
dxxfew iwxX
Illustration of the Normal Distribution
Assume that we want to model the time taken to drive a car from the NTU main campus to the NTU hospital.
According to our experience, on average, it takes 20 minutes or 1200 seconds.
Illustration of the Normal Distribution
If we left the NTU main campus and drove to the NTU hospital now, then the probability that we would arrive in 1200.333…seconds would be 0.
In addition, the probability that we would arrive in 3600.333…seconds would also be 0.
Illustration of the Normal Distribution
However, our intuition tells us that it would be more likely that we would arrive within 600 seconds and 1800 seconds than within 3000 seconds and 4200 seconds.
Illustration of the Normal Distribution
Therefore, the likelyhood function of this experiment should be of the following form:
0 600 1200 1800
Time
Likelihood
Illustration of the Normal Distribution
In fact, the probability density function models the likelihood of taking a specific amount of time to drive from the main campus to the hospital.
By the p.d.f., the probability that we would arrive within a time interval would be
dttfFF TTT
)()()(
Illustration of the Normal Distribution
In the real world, many distributions can be well modeled by the normal distributions. In other words, the profile of the p.d.f. of a normal distribution provides a good approximation of the exact p.d.f. of the distribution just like our example above.
The Standard Normal Distribution
A normal distribution with μ=0 and σ=1 is said to be a standard normal distribution.
The p.d.f. of a standard normal distribution is
.2
1 2
2
1x
e
Since is a circularly symmetric function on the Y-Z plane,
Therefore,c2=1 and c=1.
dydze
dzedyec
zy
zy
22
22
2
1
2
1
2
12
2
1
2
1
2
1
22
2
1zy
e
.22
2
02
1
0
2
1
2
1
2
222
e
dedydzezy
Linear Transformation of the Normal Distribution
Assume that random variable X has the distribution .
Then, has the standard normal distribution.
Proof:
2,N
X
Y
dxeyX
yX
yYyF
yx
Y
2
2
1
2
1Prob
ProbProb
Linear Transformation of the Normal Distribution By substituting ,
we have
Therefore, Y is Accordingly, if we want to compute , we can do that by the following procedure.
x
t
.2
1 2
2
1
dteyFty
Y
.1,0N
wF X
.ProbProbProb
wF
wY
wXwXwF YX
Expected Value and Variance of a Normal Distribution
Let X be a N(0,1).
.12
1
2
1
, Since
.2
1
][][][][
.02
1
2
1][
2222
2222
22
222
22
222
22
2
2
22
xxx
xxx
x
xx
xedxedxex
exedx
dxe
dxex
XEXEXEXVar
edxxeXE
Therefore, the expected value and variance of X are 0 and 1, respectively.
The expected value and variance of a
distribution are μ and σ2,
respectively,
since is N(0,1),
if Y is N(μ, σ2).
., 2N
Y
The Table for N( 0,1)
The Chi-Square Distribution Assume that X is N( 0,1) . In statistics, it is common that we are interested in
Therefore, we define Z=X2. The distribution function of Z is
.Prob Prob xXorxX
2
2
12
.0 ,2
1
ProbProb
z
0
2
1
2
1
0
2
1
2
22
2
dxedxe
zfordxe
zXzZzF
xxz
xz
z
Z
The Chi-Square Distribution
The p.d.f. of Z is
.0 ,2
1
where,2
2
2
1
2
1
0
2
1
0
2
1
2
2
zforez
ztdz
dt
dt
dxed
dz
dxed
dz
zdF
z
t x
z x
Z
The Chi-Square Distribution
Z is typically said to have the chi-square distribution of 1 degree of freedom and denoted by .12
Chi-Square Distribution with High Degree of Freedom
Assume that X1, X2, ……, Xk are k independent random variables and each Xi is N(0, 1).
Then, the random variable Z defined by is called a chi-square random variable with degree of freedom = k and is denoted by .
k
iiXZ
1
2
k2
Addition of Chi-Square Distributions
A
A
srsr 222
n
ii
n
ii kk
1
2
1
2
Example of Chi-Square Distribution with Degree of Freedom = 2
Assume that a computer-controlled machine is commanded to drill a hole at coordinate (10,20). The machine moves the drill along the x-axis first followed by the y-axis.
Example of Chi-Square Distribution with Degree of Freedom = 2
According to the calibration process, the positioning accuracy of the machine in terms of millimeters along each axis is governed by a normal distribution N(u,0.0625).
The engineer in charge of quality assurance determines that the center of the hole must not deviate from the expected center by more than 1.0 millimeters.
What is the defect rate of this task.
The Table of the χ2 Distribution
The distribution Function and p.d.f. of is k2
).1(11
1
t0 where
22
1
.
22
1
22
1
0
2
0
2
0
1
0
1
21
2
2
2
21
2
0 2
21
01
2
2
2
2
mmdxexm
dxexmexdxexm
etk
tf
rwwhere
dwewk
drerk
tF
xm
xmxmxm
tk
k
wkt
k
rk
t
kk
k