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Probability Distributions and Probability Densities Prob. Distributions & Densities - 1 1 Probability Distributions and Probability Densities 2 1. Random Variable Definition: If S is a sample space with a probability measure and X is a real-valued function defined over the elements of S, then X is called a random variable. [ X(s) = x] Random variables are usually denoted by capital letter. X, Y, Z, ... Lower case of letters are usually for denoting a element or a value of the random variable. 3 Discrete Random Variable Discrete Random Variable: A random variable assumes discrete values by chance. 4 Discrete Random Variable Example: (Toss a balanced coin) X = 1, if Head occurs, and X = 0, if Tail occurs. P(Head) = P(X = 1) = P(1) = .5 P(Tail) = P(X = 0) = P(0) = .5 Probability mass function: f(x) = P(X = x) =.5 , if x = 0, 1, and 0 elsewhere. 1/2 0 1 Probability Bar Chart 5 Why Random Variable? • A simple mathematical notation to describe an event. e.g.: X < 3, X = 0, ... • Mathematical function can be used to model the distribution through the use of random variable. e.g.: Binomial, Poisson, Normal, … 6 Discrete Random Variable Example: (Toss a balanced coin 3 times) X takes on number of heads occurs. Outcomes Probability x HHH 1/8 3 HHT 1/8 2 HTH 1/8 2 THH 1/8 2 HTT 1/8 1 THT 1/8 1 TTH 1/8 1 TTT 1/8 0 P(X=3) =1/8 P(X=2) =3/8 P(X=1) =3/8 P(X=0) =1/8

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Probability Distributions and Probability Densities

Prob. Distributions & Densities - 1

1

Probability Distributions and

Probability Densities

2

1. Random Variable Definition: If S is a sample space with a

probability measure and X is a real-valued

function defined over the elements of S, then

X is called a random variable. [ X(s) = x]

Random variables are usually denoted by

capital letter. X, Y, Z, ...

Lower case of letters are usually for denoting

a element or a value of the random variable.

3

Discrete Random Variable

Discrete Random Variable:

A random variable

assumes discrete values

by chance.

4

Discrete Random Variable

Example: (Toss a balanced coin)

X = 1, if Head occurs, and

X = 0, if Tail occurs.

P(Head) = P(X = 1) = P(1) = .5

P(Tail) = P(X = 0) = P(0) = .5

Probability mass function:

f(x) = P(X = x) =.5 , if x = 0, 1,

and 0 elsewhere.

1/2

0 1

Probability

Bar Chart

5

Why Random Variable?

• A simple mathematical notation to

describe an event. e.g.: X < 3, X = 0, ...

• Mathematical function can be used to

model the distribution through the use of

random variable. e.g.: Binomial, Poisson,

Normal, …

6

Discrete Random Variable Example: (Toss a balanced coin 3 times)

X takes on number of heads occurs.

Outcomes Probability x

HHH 1/8 3

HHT 1/8 2

HTH 1/8 2

THH 1/8 2

HTT 1/8 1

THT 1/8 1

TTH 1/8 1

TTT 1/8 0

P(X=3) =1/8

P(X=2) =3/8

P(X=1) =3/8

P(X=0) =1/8

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 2

7

Discrete Random Variable Example: (Toss a balanced coin 3 times)

X takes on number of heads occurs.

3,2,1,0,8

3

)()(

xx

xXPxf

Probability Distribution of X.

8

2. Probability Distribution

(or Probability Mass Function)

If X is a discrete random variable, the function

f(x) = P(X=x) is called the probability distribution

[or probability mass function (p.m.f.) ] of X, and

this function satisfies the following properties:

a) f(x) > 0, for each value of in the space S of X,

b) SxS f(x) = 1.

9

Probability Distribution

Example: Is f(x) a proper probability

distribution of the random variable X?

3. 1, 1, for ,3

)( xx

xf

10

Probability Distribution

Example: Is f(x) a proper probability

distribution of the random variable X?

3. 2, 1, for ,3

)( xx

xf

11

Probability Distribution

Example: If f(x) is a probability distribution

of a discrete random variable X, find the

value of c if

3. 2, 1, for ,

3)( xx

cxf

1)3()2()1()(3

1

fffxfi

Sol:

12

)3

3

3

2

3

1(3

32

31

3

c

cccc

2

1 c

Property 2

12

Probability Distribution

Example: Two balls are to be selected from

a box containing 5 blue balls and 3 red balls

without replacement. Find the probability

distribution for the probability of getting x

blue balls.

Sol:

2

8

1

3

1

5

)1()1( XPf

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 3

13

Probability Distribution

2. 1, 0, for ,

2

8

2

35

)(

xxx

xf

x f(x) = P(X = x)

2 20/56

1 30/56

0 15/56

14

Is it a profitable insurance

premium?

Distribution: f (-100) = .1

f (-1) = .25,

f (11) = .65,

-100 -1 11

.75

.50

.25

Probability line chart

X=x f (x)

100 .1

1 .25

11 ?

100 ,1.

1 ,25.

11 ,65.

)(

xif

xif

xif

xf

Probability Distribution

.65

15

Cumulative Distribution

Function

The cumulative distribution function (c.d.f.

or distribution function, d.f.) of a discrete

random variable is defined as

xt

xtfxXPxF .for ),()()(

• F( ) = 0, F() = 1 • If a < b, F(a) ≤ F(b) for any real numbers a, b.

16

Distribution Function Example: (Toss a balanced coin 3 times)

X takes on number of heads occurs.

x f(x) F(x)

0 1/8 1/8

1 3/8 4/8

2 3/8 7/8

3 1/8 1

17

x

F(x)

1/2

1

0 1 2 3

18

Theorem: If the range of a random variable X

consists of the values x1 < x2 < … < xn , then

f(x1) =F(x1), and f(xi) = F(xi) – F(xi-1), for i =

1, 2, 3, …

Example: (Toss a balanced coin 3 times)

X takes on number of heads occurs.

f(2) = F(2) – F(1) = 4/8 – 1/8 = 3/8

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 4

19

Sample Space (Two Dice)

1. Outcome pairs:

S = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

2. Sum: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Probability Distribution of the Sum

(Two Dice) Let X be a random variable takes on the sum of

the two dice.

P(X = 6) = ?

P(X = 8) = ?

P(X ≤ 8) = ?

20

21

Probability Distribution of the Sum

(Two Dice)

22

Distribution Function of the Sum

(Two Dice)

23

3.4 Continuous Random Variable &

Probability Density Function

Percent

A smooth curve that fit

the distribution

10 20 30 40 50 60 70 80 90 100 110

Test scores

Density

function, f (x)

f(x)

24

Definition 3.4: A function with values f(x), defined

over the set of all real numbers, is called a probability

density function (p.d.f.) of the continuous random

variable X, if and only if

f(x)

x a b

P(a ≤ X ≤ b) =

for any real constants a and b with a ≤ b.

Notes:

f(c) is the probability density at c.

f(c) ≠ P(X = c) = 0 for any continuous random variable.

𝑓 𝑥 𝑑𝑥𝑏

𝑎

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 5

25

Theorem 3.4: If X is a continuous random

variable and a and b are real constants with

a ≤ b, then

P(a ≤ X ≤ b) = P(a < X ≤ b) = P(a ≤ X 00, elsewhere

Find k.

Find P(0.5 < X < 1).

27

Definition 3.5: If X is a continuous random

variable and the value of its probability density

at t is f(t), then the function given by

𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥

= 𝑓 𝑡 𝑑𝑡, for −∞ < 𝑥 < ∞𝑥

−∞

is called the distribution function, or the

cumulative distribution of X, and

• F(−∞) = 0, F(∞) = 1, • If a < b, F(a) ≤ F(b).

28

Theorem 3.6: If f(x) and F(x) are values

of the probability density and the

distribution function of the random

variable X at x, then

P(a ≤ X ≤ b) = F(a) F(b)

for any real constants a and b with a ≤ b,

and

𝑓 𝑥 =𝑑𝐹(𝑥)

𝑑𝑥

where the derivative exists.

29

Example: If X is a continuous random

variable with a p.d.f.

𝑓 𝑥 = 3𝑒−3𝑥, for 𝑥 > 00, elsewhere

Find the distribution function of X.

Use the d.f. above to find P(0.5 < X < 1).

30

Example: Find a probability density function for the

random variable whose distribution function is given

by

𝐹 𝑥 = 0, for 𝑥 ≤ 0𝑥, for 0

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 6

31

F(x)

1

¾

½

¼

0 0.5 1

x

A Mixed Distribution

What is its F(x)?

What is its f(x)?

32

F(x)

1

½

0 1 2 3

Example: X is a random variable with the following d.f.,

Find P(X < 2) =

P(X < 1/2) =

P(X = 1) =

P(X ≤ 3/2) =

P(X > 3/2) =

𝐹 𝑥 =

0, 𝑥 < 0𝑥/2, 0 ≤ 𝑥 < 13/4, 1 ≤ 𝑥 < 25/6, 2 ≤ 𝑥 < 31, 3 ≤ 𝑥.

Multivariate Distributions

33

3.5 Multivariate Distributions

Univariate (one random variable)

Bivariate (two random variables over a joint sample space)

Multivariate (two or more random variables over a joint sample space)

If X and Y are two discrete random variables, we

denote the probability of X takes on value x and Y

takes on value y as P(X = x, Y = y).

34

Example: Two balls are selected from a box

containing 3 red balls, 2 blue balls, and 4 green balls.

If X and Y are, respectively, the numbers of red balls

and blue balls draw from the box, find the

probabilities associated with all possible pairs of

values of X and Y.

Sol: S = {(0,0), (0,1), (0,2), (1,0), (1,1), (2,0)}

6

1

2

9

2

4

0

2

0

3

)0,0(

P

35

Two balls are selected from a box containing 3 red

balls, 2 blue balls, and 4 green balls.

.20 ;2,1,0,,

2

9

2

423

),(

yxyxyxyx

yxP

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

x

Y

2

1

0

0 1 2 36

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 7

Definition 3.6. If X and Y are discrete random

variables, the function given by f(x, y) = P(X=x, Y=y)

for each pair of values (x, y) within the range of X

and Y is called the joint probability distribution of

X and Y.

37

Theorem 3.7: A bivariate function can serve the joint

probability distribution of a pair of discrete random

variables X and Y if and only if its values, f(x, y) ,

satisfy the conditions

• = 1, where the double summation

extends over all possible pairs (x, y) within its

domain.

• f(x, y) ≥ 0, for each pair of values (x, y) within

its domain; ∞ < x < ∞;

38

Example: Determine the value of k for which the

function given by f(x, y) = kxy, for x = 1, 2; and y =

1, 2, can serve as a joint probability distribution.

39

Example: In the example about drawing two

balls problem, find

• P(1 ≤ X ≤ 2, 0 ≤ Y ≤ 1) = ?

• P(X ≤ 1, 0 ≤ Y ≤ 1) = ?

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

1/6 + 1/3 + 1/12 = 7/12

2/9 + 1/6 + 1/6 + 1/3 = 7/9

40

Definition 3.7. If X and Y are discrete random

variables, the function given by

F(x, y) = P(X ≤ x, Y ≤ y) =

for ∞ < x < ∞, ∞ < y < ∞,

where f(s, t) is the value of the joint probability

distribution of X and Y at (s, t), is called the joint

distribution function, or the joint cumulative

distribution of X and Y.

xs yt tsf ),,(

41

Example: In the last example about three colors

balls problem, find

F(1, 1) = ?

F(0.5, 1.2) = ?

F(1, 1) = ?

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

x

Y

2

1

0

0 1 2

42

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 8

Definition 3.8. A bivariate function f(x, y), define

over the xy-plane, is called a joint probability

density function of the continuous random variables

X and Y if and only if

for any region A in the xy-plane.

A

dxdyyxfAYXP ),(]),[(

43

Theorem 3.8: A bivariate function can serve as a joint

probability density function of a pair of continuous

random variables X and Y if its values, f(x, y) , satisfy

the conditions

• .

• f(x, y) ≥ 0, for ∞ < x < ∞; ∞ < y < ∞,

1),( dxdyyxf

44

Example: The following is the joint probability

density function of two continuous random variables

X and Y,

elsewhere. 0

20 ,1 0for )(5

3

),(

yxyxx

yxf

1. Find P(0 < X < Y, Y < 1).

45

Example: The following is the joint probability

density function of two continuous random variables

X and Y,

elsewhere. 0

20 ,1 0for )(5

3

),(

yxyxx

yxf

2. Find P(0 < X < 1/2, 1< Y < 2).

3. Find P(0 < X + Y < 1/2).

46

Example: The following is the joint probability

density function of two continuous random variables

X and Y,

elsewhere. 0

10 , 0for 1

),(

yyxy

yxf

Find P(X + Y > 1/2).

47

y y y Definition 3.9: If X and Y are continuous random

variables with joint probability density function f(x, y),

the function given by

for ∞< x < ∞; ∞< y < ∞, is called the joint

distribution function, or the joint cumulative

distribution of X and Y.

y x

dsdttsfyYxXPyxF ),(),(),(

),(),(2

yxFyx

yxf

48

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 9

Example: If the joint probability density of X and Y

is given by

find the joint distribution function of X and Y.

49

elsewhere. 0

10 ,1 0for ),(

yxyxyxf

I II

IV III

x

y

0 1

1

50

I II

IV III

x

y

0 1

1

F(x, y)

If x < 0 or y < 0, F(x, y) = 0

If 0 < x < 1 and 0 < y < 1,

F(x, y) = y x

dsdtts0 0

)(

dttssxy

|0

2

0 21 )(

(I)

|0

2

212

21 ][

y

xttx

dtxtxy

)( 20 2

1

2

212

21 xyyx )(2

1 yxxy

51

I II

IV III

x

y

0 1

1

F(x, y)

If x > 1 and 0 < y < 1,

F(x, y) = y

dsdtts0

1

0)(

(II)

)1(21 yy

If 0 < x < 1 and y > 1,

F(x, y) = 1

0 0)(

x

dsdtts

(III)

)1(21 xx

If x > 1 and y > 1,

F(x, y) = 1

0

1

0)( dsdtts

(IV)

152

I II

IV III

x

y

0 1

1

F(x, y)

1,1for 1

1,10for )1(

10 ,1for )1(

10 ,10for )(

0,0for 0

),(

21

21

21

yx

yxxx

yxyy

yxyxxy

yx

yxF

53

Example: Find the joint probability density of the

two random variables X and Y whose joint

distribution function is given by

elsewhere 0

0 and 0for )1)(1(),(

yxeeyxF

yx

),(),(2

yxFyx

yxf

0 ,0for ,)( yxeee yxyx

and 0 elsewhere.

and also determine P(1 < X < 3, 1 < Y < 2).

Sol: For joint p.d.f.,

54

P(1 < X < 3, 1 < Y < 2) =

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 10

55

0 a b

d

c

(b, d)

(b, c)

(a, d)

(a, c)

x

y

P(a

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 11

61

3.6 Marginal Distributions

x

Y

2

1

0

0 1 2

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

5/12 1/2 1/12

1/36

7/18

7/12

62

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

5/12 1/2 1/12

The Marginal Probability Distribution of X:

2

0

.2 ,1 ,0for ),,()(y

xyxfxg

.12/1)2( ,2/1)1( ,12/5)0( ggg

63

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

1/36

7/18

7/12

The Marginal Probability Distribution of Y:

2

0

.2 ,1 ,0for ),,()(x

yyxfyh

.36/1)2( ,18/7)1( ,12/7)0( hhh

Definition 3.11. If X and Y are discrete random

variables with joint probability distribution f(x, y),

the function given by

for each x in the space of X, is called the marginal

distribution of X. Correspondingly,

64

y

yxfxg ),()(

x

yxfyh ),()(

for each y in the space of Y, is called the marginal

distribution of Y.

Definition 3.11. If X and Y are continuous random

variables with joint probability density f(x, y), the

function given by

for x < is called the marginal distribution of

X. Correspondingly,

65

dyyxfxg

),()(

for y < is called the marginal distribution of

Y.

dxyxfyh

),()(

Example: Given the joint probability density

function of two continuous random variables X and Y,

elsewhere ,0

10 ,1 0for ),2(3

2

),(yxyx

yxf

find the marginal density of X and Y.

66

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 12

Example: Given the trivariate joint probability

density function random variables X1, X2, X3,

elsewhere, ,0

0 ,10 ,1 0for ,)(),,( 32121321

3 xxxexxxxxf

x

find the marginal density of X1.

67

Example: Given the trivariate joint probability

density function random variables X1, X2, X3,

elsewhere ,0

0 ,10 ,1 0for ,)(),,( 32121321

3 xxxexxxxxf

x

find the joint marginal density of X1 and X3.

68

Definition 3.11. If X and Y are discrete random

variables with joint probability distribution f(x, y),

the function given by

for each x in the space of X, is called the marginal

distribution of X. Correspondingly,

69

y

yxfxg ),()(

x

yxfyh ),()(

for each y in the space of Y, is called the marginal

distribution of Y.

Definition 3.11. If X and Y are continuous random

variables with joint probability density f(x, y), the

function given by

for x < is called the marginal probability

density of X. Correspondingly,

70

dyyxfxg

),()(

for y < is called the marginal probability

density of Y.

dxyxfyh

),()(

71

For more than two variables, if the discrete random

variables X1, X2, X3, …, Xn has joint probability

distribution f(x1, x2, x3, …, xn), then the marginal

probability distribution of X1 is

nx

n

x

xxxfxg ),...,,(...)( 2112

for all x1 in the space of X1. The joint marginal

probability distribution of X1, X2, X3 is

nx

n

x

xxxfxxxm ),...,,(...),,( 213214

for all x1 , x2, x3, is in the space of X1, X2, X3.

72

For more than two variables, if the continuous

random variables X1, X2, X3, …, Xn has joint

probability density function f(x1, x2, x3, …, xn),

then the marginal density function of X1 is

nn dxdxxxxfxg ...),...,,(... )( 2211

for all x1 in the space of X1. The joint marginal

probability density of X1, X2, X3 is

nn dxdxxxxfxxxm ...),...,,(... ),,( 421321

for all x1 < , x2 < , …, xn < .

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 13

73

Joint Marginal Distribution Function

If F(x, y) is the joint distribution function of X

and Y, the marginal distribution function of X

is G(x) = F(x, ∞), for < x < .

If F(x1, x2, x3) is the joint distribution function of

X1, X2, X3, the joint marginal distribution

function of X1, X3, is M(x1, x3) = F(x1, ∞, x3), for

< x1 < , < x3 < .

Example: Given the joint distribution function of

two continuous random variables X and Y,

elsewhere, ,0

0 ,0for ),1)(1(),(

22

yxeeyxF

yx

find the marginal distribution of X.

74

elsewhere. ,0

0for ),1(),()(

2

xexFxG

x

x

dtdstsfxG ),()(

y x

dsdttsfyxF ),(),(

Example: Given the trivariate joint probability

density function random variables X1, X2, X3,

elsewhere, ,0

0 ,10 ,1 0for ,)(),,( 32121321

3 xxxexxxxxf

x

find the marginal distribution of X1, G(x1).

75 76

3.7 Conditional Distribution

)(

)()|(

BP

BAPBAP

Discrete Case:

If A and B are events X = x and Y = y, then

)(

),(

)(

),()|(

yh

yxf

yYP

yYxXPyYxXP

Definition 3.12. If f(x, y) is the joint probability

distribution of the discrete random variables X and

Y, and h(y) is the marginal probability distribution

of Y at y, the function given by

is called the marginal conditional distribution of X

given Y = y. Correspondingly, if g(x) is the marginal

distribution of X at x, the function given by

77

0 ,)(

),()|( h(y)

yh

yxfyxf

is called the marginal conditional distribution of Y

given X = x.

0 ,)(

),()|( xg

xg

yxfxyw

Definition 3.13. If f(x, y) is the joint p.d.f. of the

continuous random variables X and Y, and h(y) is

the marginal probability density function of Y at y,

the function given by

is called the marginal conditional probability

density function of X given Y = y. Correspondingly,

if g(x) is the marginal p.d.f. of X at x, the function

given by

78

0 ,)(

),()|( h(y)

yh

yxfyxf

is called the marginal conditional probability

density function of Y given X = x.

0 ,)(

),()|( xg

xg

yxfxyw

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 14

79

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

5/12 1/2 1/12

Conditional probability distribution of X given Y = 1:

0)1|2(

7/318/7

6/1)1|1(

7/418/7

9/2)1|0(

f

f

f

1/36

7/18

7/12

80

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

5/12 1/2 1/12

Find P( X ≤ 1 | Y = 0 )

1/36

7/18

7/12

= P( X = 0 | Y = 0 ) + P( X = 1 | Y = 0 )

)0(

)0,1(

)0(

)0 ,0(

YP

YXP

YP

YXP

7

6

12/7

3/1

12/7

6/1

81

2 1/36

y 1 2/9 1/6

0 1/6 1/3 1/12

0 1 2

x

5/12 1/2 1/12

Find P( X ≤ 1 | Y ≤ 1 ) = ?

1/36

7/18

7/12

)1(

)1 ,1(

YP

YXP

12/718/7

3/16/16/19/2

Example: Given the joint probability density

function of two continuous random variables X and Y,

elsewhere ,0

,10 ,1 0for ),2(3

2

),(yxyx

yxf

find the marginal conditional density of X given

Y = y and use it to calculate P(X ≤ 1/2 | Y = 1/2).

82

elsewhere ,0

,10 ,1 0for ),2(3

2

),(yxyx

yxf

The marginal conditional density of X given Y = y

is

83

,10for ),41(3

1)( xyyh

elsewhere. 0)|( and

,10for ,41

42

)41(3

1

)2(3

2

)(

),()|(

yxf

xy

yx

y

yx

yh

yxfyxf

84

3

22

21

22)

2

1 (

xxxf

12

5

3

22

)2

1|()

2

1 |

2

1 (

21

21

0

0

dxx

dxxfYXP

Probability Distributions and Probability Densities

Prob. Distributions & Densities - 15

85

)2

1(

)2

1 ,

2

1 (

)2

1 |

2

1 (

YP

YXP

YXP

Can we do the following if both X and Y are

continuous random variables?

)2

1(

)2

1 ,

2

1 (

)2

1 |

2

1 (

YP

YXP

YXP

Can it be done in the following way? Example: Given the joint probability density

function of two continuous random variables X and Y,

elsewhere ,0

10 ,1 0for ,4),(

yxxyyxf

find the marginal density of X and Y and the

conditional p.d.f. of X given Y = y .

86

87

More than two variables:

.0)( ,)(

),,,()|,,(

.0),( ,),(

),,,(),|,(

.0),,( ,),,(

),,,(),,|(

),,,(,,,

1

1

43211432

43

43

43214321

432

432

43214321

43214321

xlxl

xxxxfxxxxr

xxmxxm

xxxxfxxxxq

xxxgxxxg

xxxxfxxxxp

xxxxfxxxx

Definition 3.14. If f(x1, x2, x3, …, xn) is the joint

probability distribution (density) of the discrete

(continuous) random variables X1, X2, X3, …, Xn,

and fi (xi) is the marginal probability distribution

(density) of Xi , for i = 1, 2, …, n, then the n

random variables are independent if and only if

f(x1, x2, x3, …, xn) = f1(x1) f2(x2) … fn(xn).

88

Example: Consider n independent flips of a

balanced coin, let Xi be the number of heads (0 or 1)

obtained in the i-th flip for i =1,2, …, n. Find the

joint probability distribution of these n random

variables.

89

Example: Given the independent random variables

X1, X2, and X3, with the probability density functions

elsewhere, ,0

0for ,)( 111

1 xexf

x

find the joint density of X1, X2, and X3, and also

find P(X1 + X2 ≤ 1, X3 > 1). 90

elsewhere, ,0

0for ,2)( 2

2

22

2 xexf

x

elsewhere, ,0

0for ,3)( 3

3

33

3 xexf

x