Chap. 4 Continuous Distributions. Examples of Continuous Random Variable If we randomly pick up a real number between 0 and 1, then we define a continuous.

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.


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.


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.


Therefore, the likelyhood function of this experiment should be of the following form:

0 600 1200 1800

Time

Likelihood


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

)()()(


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

Chap. 4 Continuous Distributions. Examples of Continuous Random Variable If we randomly pick up a real number between 0 and 1, then we define a continuous.

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