Chapter 6Continuous Random Variables

Wen-Guey Tzeng

Computer Science Department

National Chiao Tung University

Probability density functions

Definition.• Let X be a random variable that takes values on all real

numbers • There exists a non-negative real-valued function

f: R[0, )• For an interval I=[a b], let

b

aIdxxfdxxfIXP )()()(

2

3

• X is called a continuous random variable

• The function f is called the probability density function

(pdf), or the density function of X.

• Density is not probability• f(x): we cannot say that the probability that x occurs is f(x)

• Area is probability• P(XA): the probability that X occurs in A

4

5

6

7

Distribution function for continuous random variable

• If X is a continuous random variable, then the function

𝐹 𝑡 = −∞𝑡𝑓 𝑡 𝑑𝑡

is called the (cumulative) distribution function of X.

2014 Fall 8

9

2014 Fall 10

11

F(b)-F(a)

dxxf

bXaPbXaP

bXaPbXaP

dxxfbXaP

xfxF

dxxf

dxxftF

b

a

b

a

t

)(

)()(

)()( (e)

.)()( b,a realsFor (d)

).()(' ,continuous is f If (c)

.1)( (b)

.)()( a)(

Properties

• The distribution function of a random variable X is given by

31

322/112/

212/1

104/

00

)(

x

xx

x

xx

x

xF

2014 Fall 12

• Compute the following quantities:• P(X<2)

• P(X=2)

• P(1<=X<3)

• P(X>3/2)

• P(X=5/2)

• P(2<X<=7)

2014 Fall 13

• Suppose that a bus arrives at a station every day between 10:00 A.M. and 10:30 A.M., at random.

• Let X be the arrival time; find the distribution function of X and sketch its graph.

Sol:

5.101

5.1010)10(2

00

)(

t

tt

t

tF

2014 Fall 14

• Experience has shown that while walking in a certain park, the time X, in minutes, between seeing 2 people smoking has a density function of the form

.0

0)(

otherwise

xxexf

x

15

(a) Calculate the value of (b) Find the probability distribution function of X.(c) What is the probability that Jeff, who has just seen a

person smoking, will see another person smoking in 2 to 5 minutes? In at least 7 minutes?

16

Sol:

1.or 1(-1)]-[0 Therefore

.01

lim1

lim)1(limget werule sHopital'l'By

.1])1([ Thus

.)1( parts,by n integratioBy

.)(1 )(

0

00

xxxx

x

x

x

xx

xx

ee

xex

ex

exdxxe

dxxedxxedxxfa

17

.007.0)7(1)7(

37.0)31()61()2()5()52()(

.1)1(])1([)()(

0,For t 0. tif 0)()(

25

0

FXP

eeFFXPc

etexdxxftF

tFb

ttt

x

18

• Sketch the graph of the function f(x)• Show that it is the probability density function of a

random variable X.• Find F, the distribution function of X, and show that it

is continuous.• Sketch the graph of F.

,0

51|3|4

1

2

1

)(

otherwise

xxxf

19

Density function of a function of a random variable

• f(x) is the density function of a random variable X

• What is the density function g(y) of Y=h(X)?

Two methods

• The method of distributions• Compute the distribution G(t) of Y• g(y) = G(y)’

• The method of transformations• Compute g(y) directly

20

By the method of distributions:

• Let X be a continuous random variable with the probability density function

• Find the distribution and density functions of Y=X2.

.0

21 /2)(

2

elsewhere

xifxxf

21

Sol: Let G and g be the distribution function and the density function of Y, respectively.

.2

2]2

[2

)1(

Now

.41

41)1(

10

)()()()(

11 2

2

txdx

xtXP

t

ttXP

t

tXtPtXPtYPtG

tt

22

.0

41 1/t(t)G'g(t)

:G ofation differenti

by found is g, Y, offunction density The

.41

412

2

10

)(

Therefore,

elsewhere

tift

t

tt

t

tG

23

By the method of transformations

• X is continuous with density function fX and the possible value set A.

• If Y=h(X) is invertible on B=h(A)={ h(a): a in A}.

• Then, the density function fY of Y is

B |,)()'(|))(()( 11 yyhyhfyf XY

24

Expectations and variances

Definition. If X is a continuous r. v. with the density function f(x), the expected value of X is defined by

.)()( dxxxfXE

25

• In a group of adult males, the difference between the uric acid value and 6, the standard value, is a random variable X with the following density function:

• Calculate the mean of these differences for the group.

.0

11 )32(2

1

)(2

elsewhere

xifxxxf

26

Sol:

• Remark: If X is a continuous r. v. with density function f, X is said to have a finite expected value if

.3

2]

4

3

3

2[

2

1-

)32(2

1)()(

1

1

43

1

1

32

xx

dxxxdxxxfXE

.)(|| dxxfx

27

• A r. v. X with density function f(x) is called a Cauchy random variable

(a) Find c.(b) Show that E(X) does not exist.

,x- ,1

)(2

x

cxf

28

Sol: (a)

(b)

./1 So,

.)]2

(2

[] [arctan1

1 -- 2

c

ccxcx

dxc

.)]1[ln(1

)1(2

)1(

||)(||

02

0 2- 2-

x

x

xdx

x

dxxdxxfx

29

• Let X be a continuous r. v. with density function f

• For any function h: R R

.)()()]([ dxxfxhXhE

30

• Let X be a continuous r. v. with density function f(x). Let h1, h2, …, hn be real-valued functions, and let a1, a2, …, an be real numbers. Then

)]([)]([)]([

)]()()([

2211

2211

XhEaXhEaXhEa

XhaXhaXhaE

nn

nn

31

• A point X is selected from the interval (0, /4) randomly

• Calculate E(cos2X) and E(cos2X)

Sol:

.1

2

1.

2

2

1

2

1)2cos

2

1

2

1()(cos

.2

]2sin2

[2cos4

)2(cos

6.3, Theoremby Now

.0

4/0/4)(

2

4/0

4/

0

XEXE

xxdxXE

otherwise

ttf

32

Definition. X is a continuous r. v. with density function f

• Var(X) = E(X2) - (EX)2

.)())((]))([()( 22 dxxfXExXEXEXVar

33

• The time elapsed, in minutes, between the placement of an order of pizza and its delivery is random with the density function

(a) Determine the mean and standard deviation of the time it takes for the pizza shop to deliver pizza.

(b) Suppose that it takes 12 minutes for the pizza shop to bake pizza. Determine the mean and standard deviation of the time it takes for the delivery person to deliver pizza.

.0

4025 15/1)(

otherwise

xifxf

34

• Sol:

33.4

.5.2012)()12( )(

.33.4)( hence and

,75.18)5.32(1075)( So

.107515

1)(

,5.3215

1)( )(

12-XX

X

2

40

25

22

40

25

XEXEb

XVar

XVar

dxxXE

dxxXEa

35